Asymptotic Theory of Eigenvectors for Random Matrices with Diverging Spikes

Abstract

Characterizing the asymptotic distributions of eigenvectors for large random matrices poses important challenges yet can provide useful insights into a range of statistical applications. To this end, in this paper we introduce a general framework of asymptotic theory of eigenvectors (ATE) for large spiked random matrices with diverging spikes and heterogeneous variances, and establish the asymptotic properties of the spiked eigenvectors and eigenvalues for the scenario of the generalized Wigner matrix noise. Under some mild regularity conditions, we provide the asymptotic expansions for the spiked eigenvalues and show that they are asymptotically normal after some normalization. For the spiked eigenvectors, we establish asymptotic expansions for the general linear combination and further show that it is asymptotically normal after some normalization, where the weight vector can be arbitrary. We also provide a more general asymptotic theory for the spiked eigenvectors using the bilinear form. Simulation studies verify the validity of our new theoretical results. Our family of models encompasses many popularly used ones such as the stochastic block models with or without overlapping communities for network analysis and the topic models for text analysis, and our general theory can be exploited for statistical inference in these large-scale applications.

1. Introduction

The big data era has brought us a tremendous amount of both structured and unstructured data including networks and texts in many modern applications. For network and text data, we are often interested in learning the cluster and other structural information for the underlying network communities and text topics. In these large-scale applications, we are given a network data matrix or can create such a matrix by calculating some similarity measure between text documents, where each entry of the data matrix is binary indicating the absence or presence of a link, or continuous indicating the strength of similarity between each pair of nodes or documents. Such applications naturally give rise to random matrices that can be used to reveal interesting latent structures of networks and texts for effective predictions and recommendations.

Random matrix has been widely exploited to model the interactions among the nodes of a network for applications ranging from physics and social sciences to genomics and neuroscience. Random matrix theory (RMT) has a long history and was originated by Wigner in Wigner (1955) for modeling the nucleon-nucleus interactions to understand the behavior of atomic nuclei and link the spacings of the levels of atomic nuclei to those of the eigenvalues of a random matrix. See, for example, Bai (1999) for a review of some classical technical tools such as the moment method and Stieltjes transform as well as some more recent developments on the RMT, and Mehta (2004); Tao (2004); Bai and Silverstein (2006) for detailed book-length accounts of the topic of random matrices.

There is a rich recent literature in mathematics on the asymptotic behaviors of eigenvalues and eigenvectors of random matrices (Erdős et al., 2013; Bourgade et al., 2018; Bourgade and Yau, 2017; Rudelson and Vershynin, 2016; Dekel et al., 2007). The main challenge in many RMT problems is caused by the strong dependence of eigenvalues if they are close to each other. Using the terminologies in RMT, four regimes are often of interests: bulk, subcritical edge, critical edge, and supercritical regimes. The first three regimes all have eigenvalues that are highly correlated with each other, and the last regime has weaker interactions among the eigenvalues. The last regime can be further divided into two categories according to the relative strength of spiked eigenvalues compared to noise, which can be roughly understood as the signal-to-noise ratio. There have been exciting mathematical developments in the recent mathematical literature when the smallest spiked eigenvalue has the same order as the noise (Capitaine and Donati-Martin, 2018; Knowles and Yin, 2013; Bao et al., 2018). Due to the challenge caused by constant signal-to-noise ratio, these existing results often take complicated forms and the asymptotic distributions depend generally on the noise matrix distribution in a complex way, limiting their practical usage to statisticians. In this paper, we consider the setting of diverging spikes where the spiked eigenvalues are an order of magnitude larger than the noise level asymptotically. Although mathematically easier, such random matrices are of great interests to statisticians, because many statistical applications such as network analysis and text analysis often fall into this regime. Yet there lack any formal results on the asymptotic expansions and asymptotic distributions of spiked eigenvectors even in this setting. This motivates our study in this paper.

There is a larger literature on the limiting distributions of eigenvalues than eigenvectors in RMT. For instance, the limiting spectral distribution of the Wigner matrix was generalized by Arnold (1967) and Arnold (1971). Marchenko and Pastur (1967) established the well-known Marchenko–Pastur law for the limiting spectral distribution of the sample covariance matrix including the Wishart matrix which plays an important role in statistical applications. In contrast, the asymptotic distribution of the largest nonspiked eigenvalue of Wigner matrix with Gaussian ensemble was revealed to be the Tracy–Widom law in Tracy and Widom (1994) and Tracy and Widom (1996). More recent developments on the asymptotic distribution of the largest nonspiked eigenvalue include Johnstone (2001), El Karoui (2007), Johnstone (2008), Erdös et al. (2011), and Knowles and Yin (2017). See also Füredi and Komlós (1981), Baik et al. (2005), Bai and Yao (2008), Knowles and Yin (2013), Pizzo et al. (2013), Renfrew and Soshnikov (2013), Knowles and Yin (2014), and Wang and Fan (2017) for the asymptotic distributions of the spiked eigenvalues of various random matrices and sample covariance matrices. For the eigenvectors, Capitaine and Donati-Martin (2018) and Bao et al. (2018) established their asymptotic distributions, which depend on the specific distribution of the Wigner matrix in a complicated way, in the challenging setting of constant signal-to-noise ratio. There is also a growing literature on the specific scenario and applications of large network matrices. To ensure consistency, Johnstone and Lu (2009) proposed the sparse principal component analysis to reduce the noise accumulation in high-dimensional random matrices. See, for example, McSherry (2001), Spielman and Teng (2007), Bickel and Chen (2009), Decelle et al. (2011), Rohe et al. (2011), Lei (2016), Abbe (2017), Jin et al. (2017), Chen and Lei (2018), and Vu (2018).

Matrix perturbation theory has been commonly used to characterize the deviations of empirical eigenvectors from the population ones, often under the average errors (Horn and Johnson, 2012). In contrast, recently Fan et al. (2018) and Abbe et al. (2019) investigated random matrices with low expected rank and provided a tight bound for the difference between the empirical eigenvector and some linear transformation of the population eigenvector through a delicate entrywise eigenvector analysis for the first-order approximation under the maximum norm. See also Paul (2007), Koltchinskii and Lounici (2016), Koltchinskii and Xia (2016), and Wang and Fan (2017) for the asymptotics of empirical eigenstructure for large random matrices. Yet despite these endeavors, the precise asymptotic distributions of the eigenvectors for large spiked random matrices still remain largely unknown even for the case of Wigner matrix noise. Indeed characterizing the exact asymptotic distributions of eigenvectors in such setting can provide useful insights into a range of statistical applications that involve the eigenspaces. In this sense, the asymptotic expansions and asymptotic distributions of eigenvectors established in this paper complement the existing work in the statistics literature.

The major contribution of this paper is introducing a general framework of asymptotic theory of eigenvectors (ATE) for large spiked random matrices with diverging spikes, where the mean matrix is low-rank and the noise matrix is the generalized Wigner matrix. The generalized Wigner matrix refers to a symmetric random matrix whose diagonal and upper diagonal entries are independent with zero mean, allowing for heterogeneous variances. Our family of models includes a variety of popularly used ones such as the stochastic block models with or without overlapping communities for network analysis and the topic models for text analysis. Under some mild regularity conditions, we establish the asymptotic expansions for the spiked eigenvalues and prove that they are asymptotically normal after some normalization. For the spiked eigenvectors, we provide asymptotic expansions for the general linear combination and further establish that it is asymptotically normal after some normalization for arbitrary weight vector. We also present a more general asymptotic theory for the spiked eigenvectors based on the bilinear form. To the best of our knowledge, these theoretical results are new to the literature. Our general theory can be exploited for statistical inference in a range of large-scale applications including network analysis and text analysis. For detailed comparisons with the literature, see Section 3.6.

The rest of the paper is organized as follows. Section 2 presents the model setting and theoretical setup for ATE. We establish the asymptotic expansions and asymptotic distributions for the spiked eigenvectors as well as the asymptotic distributions for the spiked eigenvalues in Section 3. Several specific statistical applications of our new asymptotic theory are discussed in Section 4. Section 5 presents some numerical examples to demonstrate our theoretical results. We further provide a more general asymptotic theory extending the results from Section 3 using the bilinear form in Section 6. Section 7 discusses some implications and extensions of our work. The proofs of main results are relegated to the Appendix. Additional technical details are provided in the Supplementary Material.

2. Model setting and theoretical setup

2.1. Model setting

As mentioned in the introduction, we focus on the class of large spiked symmetric random matrices with low-rank mean matrices and generalized Wigner matrices of noises. It is worth mentioning that our definition of the generalized Wigner matrix specified in Section 1 is broader than the conventional one in the classical RMT literature; see, for example, Yau (2012) for the formal mathematical definition with additional assumptions. To simplify the technical presentation, consider an n × n symmetric random matrix with the following structure

$$\mathbf{X}=\mathbf{H}+\mathbf{W},$$ (1)

where H = VDV^T is a deterministic latent mean matrix of low rank structure, V = (v₁, · · ·, v_K) is an n×K orthonormal matrix of population eigenvectors v_k’s with V^T V = I_K, D = diag(d₁, · · ·, d_K) is a diagonal matrix of population eigenvalues d_k’s with |d₁| ≥ ··· ≥ |d_K| > 0, and W = (w_ij)_1≤i,j≤n is a symmetric random matrix of independent noises on and above the diagonal with zero mean $\mathbb{E}{w}_{ij}=0$, variances ${\sigma }_{ij}^2=\mathbb{E}{w}_{ij}^2$, and max_1≤i,j≤n |w_ij| ≤ 1. The rank K of the mean part is assumed typically to be a smaller order of the random matrix size n, which is referred to as matrix dimensionality hereafter for convenience. The bounded assumption on w_ij is made frequently for technical simplification and satisfied in many real applications such as network analysis and text analysis. It can be relaxed to $\mathbb{E}{\left|w_{ij}\right|}^l\le C^{l-2}\mathbb{E}{\left|w_{ij}\right|}^2$, l ≥ 2, with C some positive constant, and all the proofs and results can carry through.

In practice, it is either matrix X or matrix X – diag(X) that is readily available to us, where diag(·) denotes the diagonal part of a matrix. In the context of graphs, random matrix X characterizes the connectivity structure of a graph with self loops, while random matrix X–diag(X) corresponds to a graph without self loops. In the latter case, the observed data matrix can be decomposed as

$$\mathbf{X}-\mathrm{diag}(\mathbf{X})=\mathbf{H}+[\mathbf{W}-\mathrm{diag}(\mathbf{X})].$$ (2)

Observe that W–diag(X) has the similar structure as W in the sense of being symmetric and having bounded independent entries on and above the diagonal, by assuming that diag(X) has bounded entries for such a case. Thus models (1) and (2) share the same decomposition of a deterministic low rank matrix plus some symmetric noise matrix of bounded entries, which is roughly all we need for the theoretical framework and technical analysis. For these reasons, to simplify the technical presentation we abuse slightly the notation by using X and W to represent the observed data matrix and the latent noise matrix, respectively, in either model (1) or model (2). Therefore, throughout the paper the data matrix X may have diagonal entries all equal to zero and correspondingly the noise matrix W may have a nonzero diagonal mean matrix, and our theory covers both cases.

In either of the two scenarios discussed above, we are interested in inferring the structural information in models (1) and (2), which often boils down to the latent eigenstructure (D,V). Since both the eigenvector matrix V and eigenvalue matrix D are unavailable to us, we resort to the observable random data matrix X for extracting the structural information. To this end, we conduct a spectral decomposition of X, and denote by λ₁, · · ·, λ_n its eigenvalues and ${\widehat{\mathbf{v}}}_1,\cdots ,{\widehat{\mathbf{v}}}_n$ the corresponding eigenvectors. Without loss of generality, assume that |λ₁| ≥ ··· ≥ |λ_n| and denote by $\widehat{\mathbf{V}}=\left({\widehat{\mathbf{v}}}_1,\cdots ,{\widehat{\mathbf{v}}}_K\right)$ an n × K matrix of spiked eigenvectors. As mentioned before, we aim at investigating the precise asymptotic behavior of the spiked empirical eigenvalues λ₁, · · ·, λ_K and spiked empirical eigenvectors ${\widehat{\mathbf{v}}}_1,\cdots ,{\widehat{\mathbf{v}}}_K$ of data matrix X. It is worth mentioning that our definition of spikedness differs from the conventional one in that the underlying rank order depends on the magnitude of eigenvalues instead of the nonnegative eigenvalues that are usually assumed.

One concrete example is the stochastic block model (SBM), where the latent mean matrix H takes the form H = ΠPΠ^T with $\mathbf{\Pi }={\left({\pi }_1,\cdots ,{\pi }_n\right)}^T\in {ℝ}^{n\times K}$ a matrix of community membership vectors and $\mathbf{P}=\left(p_{kl}\right)\in {ℝ}^{K\times K}$ a nonsingular matrix with p_kl ∈ [0, 1] for 1 ≤ k, l ≤ K. Here, for each 1 ≤ i ≤ n, π_i ∈ {e₁, · · ·, e_K} with ${\mathbf{e}}_j\in {ℝ}^K$, 1 ≤ j ≤ K, a unit vector with the kth component being one and all other components being zero. It is well known that the community information of the SBM is encoded completely in the eigenstructure of the mean matrix H, which serves as one of our motivations for investigating the precise asymptotic distributions of the empirical eigenvectors and eigenvalues.

2.2. Theoretical setup

We first introduce some notation that will be used throughout the paper. We use a ≪ b to represent a/b → 0 as matrix size n increases. We say that an event ${\mathcal{E}}_n$ holds with significant probability if $\mathbb{P}\left({\mathcal{E}}_n\right)=1-O\left(n^{-l}\right)$ for some positive constant l and sufficiently large n. For a matrix A, we use λ_j(A) to denote the jth largest eigenvalue in magnitude, and ‖A‖_F, ‖A‖, and ‖A‖_∞ to denote the Frobenius norm, the spectral norm, and the matrix entrywise maximum norm, respectively. Denote by A_−k the submatrix of A formed by removing the kth column. For any n-dimensional unit vector x = (x₁, · · ·, x_n)^T, let d_x = ‖x‖_∞ represent the maximum norm of the vector.

We next introduce a definition that plays a key role in proving all asymptotic normality results in this paper.

Definition 1.

A pair of unit vectors (x, y) of appropriate dimensions is said to satisfy the W^l-CLT condition for some positive integer l if ${\mathbf{x}}^T\left({\mathbf{W}}^l-\mathbb{E}{\mathbf{W}}^l\right)\mathbf{y}$ is asymptotically standard normal after some normalization, where CLT refers to the central limit theorem.

Lemmas 1 and 2 below provide some sufficient conditions under which (x, y) can satisfy the W^l-CLT condition defined in Definition 1 for l = 1 and 2, which is all we need for our technical analysis of asymptotic distributions. In this paper, we apply these lemmas with either x or y equal to v_k. Therefore, a sufficient condition for the results in our paper is that ‖v_k‖_∞ is small enough.

Lemma 1.

Assume that n-dimensional unit vectors x and y satisfy

$$\Vert \mathbf{x}{\Vert }_{\infty }\Vert \mathbf{y}{\Vert }_{\infty }\ll {\left[\mathrm{var}\left({\mathbf{x}}^T\mathbf{W}\text{y}\right)\right]}^{1/2}=s_n.$$ (3)

Then x^T Wy satisfies the Lyapunov condition for CLT and we have $\left({\mathbf{x}}^T\mathbf{W}\text{y}-\mathbb{E}{\mathbf{x}}^T\mathbf{W}\text{y}\right)/s_n\stackrel{\mathcal{D}}{\to }N\text{(0,1)}$ as n → ∞, which entails that (x, y) satisfies the W^l-CLT condition with l = 1.

To introduce W²-CLT, for any given unit vectors x = (x₁, · · ·, x_n)^T and y = (y₁, · · ·, y_n)^T, we denote respectively _{$s_{\mathbf{x},\mathbf{y}}^2$} and κ_x,y the mean and variance of the random variable

$$\begin{gathered} \sum _{1\le k,i\le n,k\le i}{\sigma }_{ki}^2[\sum _{1\le l<k\le n}{w}_{il}\left(x_k{y}_l+y_k{x}_l\right)+\sum _{1\le l<i\le n}{w}_{kl}\left(x_i{y}_l+y_i{x}_l\right) \\ {+\left(1-{\delta }_{ki}\right)\mathbb{E}{w}_{ii}\left(x_i{y}_k+x_k{y}_i\right)]}^2+2\sum _{1\le k,i\le n,k\le i}{\gamma }_{ki}\left(x_k{y}_k+x_i{y}_i\right) \\ \times [\sum _{1\le l<k\le n}{w}_{il}\left(x_k{y}_l+y_k{x}_l\right)+\sum _{1\le l<i\le n}{w}_{kl}\left(x_i{y}_l+y_i{x}_l\right) \\ +\left(1-{\delta }_{ki}\right)\mathbb{E}{w}_{ii}\left(x_i{y}_k+x_k{y}_i\right)]+\sum _{1\le k,i\le n,k\le i}{\kappa }_{ki}{\left(x_k{y}_k+x_i{y}_i\right)}^2, \end{gathered}$$ (4)

where ${\gamma }_{ki}=\mathbb{E}{w}_{ki}^3$ and ${\kappa }_{ki}=\mathbb{E}{\left(w_{ki}^2-{\sigma }_{ki}^2\right)}^2$ for k ≠ i, ${\gamma }_{kk}=2\left(\mathbb{E}{\omega }_{kk}^3-{\sigma }_{kk}^2\mathbb{E}{\omega }_{kk}\right)$, ${\kappa }_{kk}=4\mathbb{E}{\left({\omega }_{kk}^2-{\sigma }_{kk}^2\right)}^2$ with ω_kk = 2⁻¹w_kk, ${\sigma }_{kk}^2=\mathbb{E}{\omega }_{kk}^2$, and δ_ki = 1 when k = i and 0 otherwise. It is worth mentioning that the random variable given in (4) coincides with the one defined in (A.7) in Section B.2 of Supplementary Material, which is simply the conditional variance of random variable ${\mathbf{x}}^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right)\mathbf{y}$ given in (A.5) when expressed as a sum of martingale differences with respect to a suitably defined σ-algebra; see Section B.2 for more technical details and the precise expressions for $s_{\mathbf{x},\mathbf{y}}^2$ and κ_x,y given in (A.8) and (A.9), respectively.

Lemma 2.

Assume that n-dimensional unit vectors x and y satisfy $\Vert \mathbf{x}{\Vert }_{\infty }\Vert \mathbf{y}{\Vert }_{\infty }\to 0,{\kappa }_{\mathbf{x},\mathbf{y}}^{1/4}\ll s_{\mathbf{x},\mathbf{y}}$, and s_x,y → ∞. Then we have $\left[{\mathbf{x}}^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right)\mathbf{y}\right]/s_{\mathbf{x},\mathbf{y}}\stackrel{\mathcal{D}}{\to }N(0,1)$ as n → ∞, which entails that (x, y) satisfies the W²-CLT condition.

Remark 1.

To provide more insights into the conditions of Lemmas 1 and 2, we discuss the special case of standard Wigner matrix where ${\sigma }_{ij}^2=p(1-p)$ with p the expected value of entries of X. Then $s_n^2:=\mathrm{var}\left({\mathbf{x}}^T\mathbf{W}\text{y}\right)\in [p(1-p),2p(1-p)]$ and condition (3) in Lemma 1 reduces to

$$\Vert \mathbf{x}{\Vert }_{\infty }\Vert \mathbf{y}{\Vert }_{\infty }\ll \sqrt{\mathrm{var}(\mathbf{x}W\text{y})}~\sqrt{p(1-p)}.$$

Moreover, (A.13) in the Supplementary Material ensures that Lemma 2 holds under the following sufficient conditions

$$\Vert \mathbf{x}{\Vert }_{\infty }\Vert \mathbf{y}{\Vert }_{\infty }\to 0,n^{3/2}p(1-p)\Vert \mathbf{x}{\Vert }_{\infty }^2\Vert \mathbf{y}{\Vert }_{\infty }^2\to 0,andp(1-p)n\to \infty .$$ (5)

Thus if either ‖x‖_∞ or ‖y‖_∞ is small enough, both lemmas hold. Indeed in this scenario, direct calculations show that ${\mathbf{s}}_{\text{x},\text{y}}^2~np(1-p)$.

We see from Lemmas 1 and 2 that the W^l-CLT condition defined in Definition 1 can indeed be satisfied under some mild regularity conditions. In particular, Definition 1 is important to our technical analysis since to establish the asymptotic normality of the spiked eigenvectors and spiked eigenvalues, we first need to expand the target to the form of ${\mathbf{x}}^T\left({\mathbf{W}}^l-\mathbb{E}{\mathbf{W}}^l\right)\text{y}$ with l some positive integer plus some small order term, and then the asymptotic normality follows naturally if (x, y) satisfies the W^l-CLT condition. To facilitate our technical presentation, let us introduce some further notation. For any t ≠ 0 and given matrices M₁ and M₂ of appropriate dimensions, we define the function

$$\mathcal{R}\left({\mathbf{M}}_1,{\mathbf{M}}_2,t\right)=-\sum _{l=0,l\ne 1}^L{t}^{-(l+1)}{\mathbf{M}}_1^T\mathbb{E}{\mathbf{W}}^l{\mathbf{M}}_2,$$ (6)

where L is some sufficiently large positive integer that will be specified later in our technical analysis. For each 1 ≤ k ≤ K, any given matrices M₁ and M₂ of appropriate dimensions, and n-dimensional vector u, we further define functions

$$\mathcal{P}\left({\mathbf{M}}_1,{\mathbf{M}}_2,t\right)=t\mathcal{R}\left({\mathbf{M}}_1,{\mathbf{M}}_2,t\right),{\tilde{\mathcal{P}}}_{k,t}={\left[t^2{\left(A_{{\mathbf{v}}_k,k,t}/t\right)}^{\prime }\right]}^{-1},$$ (7)

$${\mathbf{b}}_{\mathbf{u},k,t}=\mathbf{u}-{\mathbf{V}}_{-k}{\left[{\left({\mathbf{D}}_{-k}\right)}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t\right)\right]}^{-1}\mathcal{R}{\left(\mathbf{u},{\mathbf{V}}_{-k},t\right)}^T,$$ (8)

where D_−k denotes the submatrix of the diagonal matrix D by removing the kth row and kth column,

$$A_{\mathbf{u},k,t}=\mathcal{P}\left(\mathbf{u},{\mathbf{v}}_k,t\right)-\mathcal{P}\left(\mathbf{u},{\mathbf{V}}_{-k},t\right){\left[t{\left({\mathbf{D}}_{-k}\right)}^{-1}+\mathcal{P}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t\right)\right]}^{-1}\mathcal{P}\left({\mathbf{V}}_{-k},{\mathbf{v}}_k,t\right),$$ (9)

(·)′ denotes the derivative with respect to scalar t or complex variable z throughout the paper, and the rest of notation is the same as introduced before.

3. Asymptotic distributions of spiked eigenvectors

3.1. Technical conditions

To facilitate our technical analysis, we need some basic regularity conditions.

Condition 1.

Assume that ${\alpha }_n={\Vert \mathbb{E}{(\mathbf{W}-\mathbb{E}\mathbf{W})}^2\Vert }^{1/2}\to \infty$ as n → ∞.

Condition 2.

There exists a positive constant c₀ < 1 such that min{|d_i|/|d_j| : 1 ≤ i < j ≤ K + 1, d_i ≠ −d_j} ≥ 1 + c₀. In addition, either of the following two conditions holds:

1. |d_K|=(n^ϵα_n) → ∞ with some small positive constant ϵ,

2. ${\max }_{i,j}\mathrm{var}\left(w_{ij}\right)\le \left(c_1^2{\alpha }_n^2\right)/n$ and |d_K| > cα_n log n with some constants c₁ ≥ 1 and c > 4c₁(1 + 2⁻¹c₀).

Condition 3.

It holds that |d₁| = O(|d_K|), |d_K|σ_min/α_n → ∞, ${\Vert {\mathbf{v}}_k\Vert }_{\infty }^2/{\sigma }_{\min }\to 0$, ${\alpha }_n^4{\Vert {\mathbf{v}}_k\Vert }_{\infty }^4/\left(\sqrt{n}{\sigma }_{\min }^2\right)\to 0$, and ${\sigma }_{\min }^{2}n\to \infty$ , where ${\sigma }_{\min }={\left\{{\min }_{1\le i,j\le n,i\ne j}\mathbb{E}{w}_{ij}^2\right\}}^{1/2}$.

Conditions 1–2 are needed in all our Theorems 1–5 and imposed for our general model (1), including the specific case of sparse models. In contrast, condition 3 is required only for Theorem 3 under some specific models with dense structures such as the stochastic block models with or without overlapping communities.

Condition 1 restricts essentially the sparsity level of the random matrix (e.g., given by a network). Note that it follows easily from max_1≤i,j≤n |w_ij| ≤ 1 that α_n ≤ n^1/2. It is a rather mild condition that can be satisfied by very sparse networks. For example, if $\mathbb{E}{w}_{12}^2=\cdots =\mathbb{E}{w}_{1\lfloor \log n\rfloor }^2=1/2$ and the other w_1j’s are equal to zero, then we have ${\alpha }_n^2\ge 2^{-1}\log n\to \infty$. Many network models in the literature satisfy this condition; see, for example, Jin et al. (2017), Lei (2016), and Zhang et al. (2015).

Condition 2 requires that the spiked population eigenvalues of the mean matrix H (in the diagonal matrix D) are simple and there is enough gap between the eigenvalues. The constant c₀ can be replaced by some o(1) term and our theoretical results can still be proved with more delicate derivations. This requirement ensures that we can obtain higher order expansions of the general linear combination for each empirical eigenvector precisely. Otherwise if there exist some eigenvalues such that d_i = d_i+1, then ${\widehat{\mathbf{v}}}_i$ and ${\widehat{\mathbf{v}}}_{i+1}$ are generally no longer identifiable so we cannot derive clear asymptotic expansions for them; see also Abbe et al. (2019) for related discussions. Condition 2 also requires a gap between α_n and |d_K|. Since parameter α_n reflects the strength of the noise matrix W, it requires essentially the signal part H to dominate the noise part W with some asymptotic rate. Similar condition is used commonly in the network literature; see, for instance, Abbe et al. (2019) and Jin et al. (2017).

Condition 3 restricts our attention to some specific dense network models. In particular, |d₁| = O(|d_K|) assumes that the eigenvalues in D share the same order. The other assumptions in Condition 3 require essentially that the minimum variance of the off-diagonal entries of W cannot tend to zero too fast, which is used only to establish a more simplified theory under the more restrictive model; see Theorem 3.

3.2. Asymptotic distributions of spiked eigenvalues

We first present the asymptotic expansions and CLT for the spiked empirical eigenvalues λ₁, · · ·, λ_K. For each 1 ≤ k ≤ K, denote by t_k the solution to equation

$$\begin{gathered} f_k(z)=1+d_k\{\mathcal{R}\left({\mathbf{v}}_k,{\mathbf{v}}_k,z\right)-\mathcal{R}\left({\mathbf{v}}_k,{\mathbf{V}}_{-k},z\right){\left[{\left({\mathbf{D}}_{-k}\right)}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1} \\ \times \mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{v}}_k,z\right)\}=0 \end{gathered}$$ (10)

when restricted to the interval z ∈ [a_k, b_k], where

$$a_k=\{\begin{array}{ll}{d}_k/\left(1+2^{-1}{c}_0\right),\hfill & d_k>0\hfill \\ \left(1+2^{-1}{c}_0\right)d_k,\hfill & d_k<0\hfill \end{array}\text{and}{b}_k=\{\begin{array}{ll}\left(1+2^{-1}{c}_0\right)d_k,\hfill & d_k>0\hfill \\ d_k/\left(1+2^{-1}{c}_0\right),\hfill & d_k<0\hfill \end{array}.$$

The following lemma characterizes the properties of the population quantity t_k’s defined in (10): It is unique and the asymptotic mean of λ_k.

Lemma 3.

Equation (10) has a unique solution in the interval z ∈ [a_k, b_k] and thus t_k’s are well defined. Moreover, for each 1 ≤ k ≤ K we have t_k/d_k → 1 as n → ∞.

It is seen from Lemma 3 that when the matrix size n is large enough, the values of t_k and d_k are very close to each other. The following theorem establishes the asymptotic expansions and CLT for λ_k and reveals that t_k is in fact its asymptotic mean.

Theorem 1.

Under Conditions 1–2, for each 1 ≤ k ≤ K we have

$${\lambda }_k-t_k={\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n{d}_k^{-1}\right).$$ (11)

Moreover, if $\mathrm{var}\left({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\right)\gg {\alpha }_n^2{d}_k^{-2}$ and the pair of vectors (v_k, v_k) satisfies the W¹-CLT condition, then we have

$$\frac{{\lambda }_k-t_k-\mathbb{E}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k}{{\left[\mathrm{var}\left({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\right)\right]}^{1/2}}\stackrel{\mathcal{D}}{\to }N(0,1).$$ (12)

Capitaine et al. (2012) and Knowles and Yin (2014) established the joint distribution of the spiked eigenvalues for the deformed Wigner matrix in different settings than ours. Capitaine et al. (2012) assumed that $\mathbb{E}{w}_{ii}^2=1/2$ and $\mathbb{E}{w}_{ij}^2=1$ for i ≠ j, while Knowles and Yin (2014) assumed that $\mathbb{E}{w}_{ij}^2=1$ for all (I,j). Under their model settings, the smallest spiked eigenvalue |d_K| and the noise level α_n are of the same order, and as a result, their asymptotic distributions depend on the distributions of the Wigner matrix. In contrast, our Theorem 1 is proved in the setting of diverging spikes. Thanks to the stronger signal-to-noise ratio, the noise matrix contributes to the distributions of the spiked eigenvalues in Theorem 1 in a global way, allowing for more heterogeneity in the variances of entries of the noise matrix W.

Theorem 1 requires that (v_k, v_k) satisfies the W¹-CLT condition and $\mathrm{var}\left({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\right)\gg {\alpha }_n^2{d}_k^{-2}$. To gain some insights into these two conditions, we will provide some sufficient conditions for such assumptions. Let us consider the specific case of σ_min > 0, that is, the generalized Wigner matrix W is nonsparse. We will show that as long as

$${\Vert {\mathbf{v}}_k\Vert }_{\infty }^2{\sigma }_{\min }^{-1}\to 0\text{and}{\sigma }_{\min }\gg {\alpha }_n{\left|d_k\right|}^{-1},$$ (13)

the aforementioned two conditions in Theorem 1 hold. We first verify the W¹-CLT condition. By Lemma 1, a sufficient condition for (v_k, v_k) to satisfy the W¹-CLT condition is that

$${\Vert {\mathbf{v}}_k\Vert }_{\infty }^2\ll {\left[\mathrm{var}\left({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\right)\right]}^{1/2}={\left[\mathbb{E}{\left({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k-\mathbb{E}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\right)}^2\right]}^{1/2}.$$ (14)

Observe that it follows from ${\sum }_{1\le i\le n}{\left({\mathbf{v}}_k\right)}_i^2={\mathbf{v}}_k^T{\mathbf{v}}_k=1$ and ${\sum }_{1\le i\le n}{\left({\mathbf{v}}_k\right)}_i^4\le {\Vert {\mathbf{v}}_k\Vert }_{\infty }^2\le 1$ that

$$\begin{gathered} {\left[\mathbb{E}{\left({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k-\mathbb{E}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\right)}^2\right]}^{1/2}\ge {\left[2\sum _{1\le i,j\le n,i\ne j}{\sigma }_{ij}^2{\left({\mathbf{v}}_k\right)}_i^2{\left({\mathbf{v}}_k\right)}_j^2\right]}^{1/2} \\ \ge {\sigma }_{\min }{\left[2\sum _{1\le i,j\le n,i\ne j}{\left({\mathbf{v}}_k\right)}_i^2{\left({\mathbf{v}}_k\right)}_j^2\right]}^{1/2}={\sigma }_{\min }{\left[2-2\sum _{1\le i\le n}{\left({\mathbf{v}}_k\right)}_i^4\right]}^{1/2} \\ \ge {\sigma }_{\min }{\left(2-2{\Vert {\mathbf{v}}_k\Vert }_{\infty }^2\right)}^{1/2}, \end{gathered}$$ (15)

where (v_k)_i stands for the ith component of vector v_k. The assumption ${\Vert {\mathbf{v}}_k\Vert }_{\infty }^2{\sigma }_{\min }^{-1}\to 0$ in (13) together with (15) ensures (14), which consequently entails that (v_k, v_k) satisfies the W¹-CLT condition.

We next check the condition $\mathrm{var}\left({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\right)=\mathbb{E}{\left({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k-\mathbb{E}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\right)}^2\gg {\alpha }_n^2{d}_k^{-2}$. It follows directly from (15) that this condition holds under (13). In fact, since Condition 2 guarantees that α_n/|d_k| asymptotically vanishes, the assumption ${\sigma }_{\min }\gg {\alpha }_n{\left|d_k\right|}^{-1}$ can be very mild. In particular, for the Wigner matrix W with σ_ij ≡ 1 for all 1 ≤ i, j ≤ n, it holds that

$$\mathbb{E}{\left({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k-\mathbb{E}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\right)}^2=2.$$ (16)

Thus the condition of $\mathbb{E}{\left({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k-\mathbb{E}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\right)}^2\gg {\alpha }_n^2{d}_k^{-2}$ reduces to that of ${\alpha }_n^2{d}_k^{-2}\ll 1$, which is guaranteed to hold under Condition 2.

We also would like to point out that one potential application of the new results in Theorem 1 is determining the number of spiked eigenvalues, which in the network models reduces to determining the number of non-overlapping (or possibly overlapping) communities or clusters.

3.3. Asymptotic distributions of spiked eigenvectors

We now present the asymptotic distributions of the spiked empirical eigenvectors ${\widehat{\mathbf{v}}}_k$ for 1 ≤ k ≤ K. To this end, we will first establish the asymptotic expansions and CLT for the bilinear form

$${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}$$

with 1 ≤ k ≤ K, where $\mathbf{x},\mathbf{y}\in {ℝ}^n$ are two arbitrary non-random unit vectors. Then by setting y = v_k, we can establish the asymptotic expansions and CLT for the general linear combination ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k$. Although the limiting distribution of the bilinear form ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}$ is the theoretical foundation for establishing the limiting distribution of the general linear combination ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k$, due to the technical complexities we will defer the theorems summarizing the limiting distribution of ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}$ to a later technical section (i.e., Section 6), and present only the results for ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k$ in this section. This should not harm the flow of the paper. For readers who are also interested in our technical proofs, they can refer to Section 6 for more technical details; otherwise it is safe to skip that technical section. For each 1 ≤ k ≤ K, let us choose the direction of ${\widehat{\mathbf{v}}}_k$ such that ${\mathbf{v}}_k^T{\widehat{\mathbf{v}}}_k\ge 0$ for the theoretical derivations, which is always possible after a sign change when needed.

Theorem 2.

Under Conditions 1–2, for each 1 ≤ k ≤ K we have the following properties:

1. If the unit vector u satisfies that |u^T v_k| ∈ [0, 1) and ${\alpha }_n^{-2}{d}_k^2\mathrm{var}\left[\left({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T\right)\mathbf{W}{\mathbf{v}}_k\right]\to \infty$, then it holds that

   $$\begin{gathered} t_k\left({\mathbf{u}}^T{\widehat{\mathbf{v}}}_k+A_{\mathbf{u},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}^{1/2}\right)=\left({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T\right)\mathbf{W}{\mathbf{v}}_k \\ +o_p\left({\left\{\mathrm{var}\left[({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k\right]\right\}}^{1/2}\right), \end{gathered}$$ (17)

   where the asymptotic mean has the expansion $A_{\mathbf{u},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}^{1/2}=-{\mathbf{u}}^T{\mathbf{v}}_k+O\left({\alpha }_n^2{d}_k^{-2}\right)$. Furthermore, if $\left({\mathbf{b}}_{\mathbf{u},k,t_k}-{\mathbf{v}}_k{\mathbf{v}}_k^T\mathbf{u},{\mathbf{v}}_k\right)$ satisfies the W¹-CLT condition, then it holds that

   $$\frac{t_k\left({\mathbf{u}}^T{\widehat{\mathbf{v}}}_k+A_{\mathbf{u},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}^{1/2}\right)-\mathbb{E}\left[\left({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T\right)\mathbf{W}{\mathbf{v}}_k\right]}{{\left\{\mathrm{var}\left[({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k\right]\right\}}^{1/2}}\stackrel{\mathcal{D}}{\to }N(0,1).$$
2. If $\left({\alpha }_n^{-4}{d}_k^2+1\right)\mathrm{var}\left({\mathbf{v}}_k^T{\mathbf{W}}^2{\mathbf{v}}_k\right)\to \infty$, then it holds that

   $$2t_k^2\left({\mathbf{v}}_k^T{\widehat{\mathbf{v}}}_k+A_{{\mathbf{v}}_k,k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}^{1/2}\right)=-{\mathbf{v}}_k^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k+o_p\left\{{\left[\mathrm{var}\left({\mathbf{v}}_k^T{\mathbf{W}}^2{\mathbf{v}}_k\right)\right]}^{1/2}\right\},$$ (18)

   where the asymptotic mean has the expansion $A_{{\mathbf{v}}_k,k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}^{1/2}=-1+2^{-1}{t}_k^{-2}{\mathbf{v}}_k^T\mathbb{E}{\mathbf{W}}^2{\mathbf{v}}_k+O\left({\alpha }_n^3{d}_k^{-3}\right)$. Furthermore, if (v_k, v_k) satisfies the W²-CLT condition, then it holds that

   $$\frac{2t_k^2\left({\mathbf{v}}_k^T{\widehat{\mathbf{v}}}_k+A_{{\mathbf{v}}_k,k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}^{1/2}\right)}{{\left[\mathrm{var}\left({\mathbf{v}}_k^T{\mathbf{W}}^2{\mathbf{v}}_k\right)\right]}^{1/2}}\stackrel{\mathcal{D}}{\to }N(0,1).$$

The two parts of Theorem 2 correspond to two different cases when $\mathrm{var}\left({\mathbf{u}}^T{\widehat{\mathbf{v}}}_k\right)$ can be of different magnitude. To understand this, note that for large enough matrix size n, we have |t_K| ≫ α_n by Condition 2 and Lemma 3. In view of (18), the asymptotic variance of ${\mathbf{v}}_k^T{\widehat{\mathbf{v}}}_k$ is equal to $\mathrm{var}(2^{-1}{t}_k^{-2}{\mathbf{v}}_k^T{\mathbf{W}}^2{\mathbf{v}}_k)$. In contrast, in light of (17), the asymptotic variance of ${\mathbf{u}}^T{\widehat{\mathbf{v}}}_k$ with |u^T v_k| ∈ [0, 1) is equal to $\mathrm{var}\left[t_k^{-1}({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k\right]$. Let us consider a specific case when $\mathrm{var}\left[({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k\right]~1$. By Lemma 4 in Section 6, we have

$$\mathrm{var}\left(2^{-1}{t}_k^{-2}{\mathbf{v}}_k^T{\mathbf{W}}^2{\mathbf{v}}_k\right)=O\left({\alpha }_n^2{t}_k^{-4}\right)\ll \mathrm{var}\left[t_k^{-1}\left({\mathbf{b}}_{u,k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T\right)\mathbf{W}{\mathbf{v}}_k\right]=O\left(t_k^{-2}\right).$$

This shows that the above two cases can be very different in the magnitude for the asymptotic variance of ${\mathbf{u}}^T{\widehat{\mathbf{v}}}_k$ and thus should be analyzed separately.

To gain some insights into why ${\mathbf{v}}_k^T{\widehat{\mathbf{v}}}_k$ has smaller variance, let us consider the simple case of K = 1. Then in view of our technical arguments, it holds that

$$\begin{gathered} {\mathbf{v}}_1^T{\widehat{\mathbf{v}}}_1{\widehat{\mathbf{v}}}_1^T{\mathbf{v}}_1=-\frac{1}{2\pi i}{\oint }_{{\Omega }_1}\frac{{\mathbf{v}}_1^T{(\mathbf{W}-z\mathbf{I})}^{-1}{\mathbf{v}}_1}{1+d_1{\mathbf{v}}_1^T{(\mathbf{W}-z\mathbf{I})}^{-1}{\mathbf{v}}_1}dz \\ =-\frac{1}{2\pi i}{\oint }_{{\Omega }_1}\frac{1}{{\left[{\mathbf{v}}_1^T{(\mathbf{W}-z\mathbf{I})}^{-1}{\mathbf{v}}_1\right]}^{-1}+d_1}dz, \end{gathered}$$ (19)

where $i=\sqrt{-1}$ is associated with the complex integrals represents the imaginary unit and the line integrals are taken over the contour Ω₁ that is centered at (a₁ + b₁)/2 with radius c₀|d₁|/2. Then we can see that the population eigenvalue d₁ is enclosed by the contour Ω₁. By the Taylor expansion, we can show that with significant probability,

$${\left[{\mathbf{v}}_1^T{(\mathbf{W}-z\mathbf{I})}^{-1}{\mathbf{v}}_1\right]}^{-1}=-z-{\mathbf{v}}_1^T\mathbf{W}{\mathbf{v}}_1+O\left(|z{|}^{-1}{\alpha }_n^2\log n\right).$$

Substituting the above expansion into (19) results in

$$\begin{gathered} -\frac{1}{2\pi i}{\oint }_{{\Omega }_1}\frac{1}{{\left[{\mathbf{v}}_1^T{(\mathbf{W}-z\mathbf{I})}^{-1}{\mathbf{v}}_1\right]}^{-1}+d_1}dz=-\frac{1}{2\pi i}{\oint }_{{\Omega }_1}\frac{1}{d_1-z-{\mathbf{v}}_1^T\mathbf{W}{\mathbf{v}}_1+O\left(|z{|}^{-1}{\alpha }_n^2\log n\right)}dz \\ =-\frac{1}{2\pi i}{\oint }_{{\Omega }_1}\frac{1}{d_1-z}-\frac{1}{2\pi i}{\oint }_{{\Omega }_1}\frac{{\mathbf{v}}_1^T\mathbf{W}{\mathbf{v}}_1}{{\left(d_1-z\right)}^2}dz+O\left(d_1^{-2}{\alpha }_n^2\log n\right) \\ =-\frac{1}{2\pi i}{\oint }_{{\Omega }_1}\frac{1}{d_1-z}+O\left(d_1^{-2}{\alpha }_n^2\log n\right) \end{gathered}$$ (20)

with significant probability. Thus the asymptotic distribution of ${\mathbf{v}}_1^T{\widehat{\mathbf{v}}}_1{\widehat{\mathbf{v}}}_1^T{\mathbf{v}}_1$ is determined by $O_p\left(d_1^{-2}{\alpha }_n^2\log n\right)$, which has no contribution from ${\mathbf{v}}_1^T\mathbf{W}{\mathbf{v}}_1$. On the other hand, our technical analysis for ${\mathbf{u}}^T{\widehat{\mathbf{v}}}_1{\widehat{\mathbf{v}}}_1{\mathbf{v}}_1$ (which is much more complicated and can be found in the technical proofs section) reveals that the dominating term is u^TWv₁ when u ≠ v₁ or −v₁. This explains why we need to treat differently the two cases of u close to or far away from v₁.

3.4. A more specific structure and an application

Theorem 2 in Section 3.3 provides some general sufficient conditions to ensure the asymptotic normality for the spiked empirical eigenvectors. Under some simplified but stronger assumptions in Condition 3, the same results on the empirical eigenvectors and eigenvalues continue to hold. Note that the stochastic block models with non-overlapping or overlapping communities can both be included as specific cases of our theoretical analysis. As mentioned before, we choose the direction of ${\widehat{\mathbf{v}}}_k$ such that ${\mathbf{v}}_k^T{\widehat{\mathbf{v}}}_k\ge 0$ for each 1 ≤ k ≤ K.

Theorem 3.

Under Conditions 1–3, for each 1 ≤ k ≤ K we have the following properties:

1. (Eigenvalues) It holds that

   $$\frac{{\lambda }_k-t_k-\mathbb{E}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k}{{\left[\mathbb{E}{({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k-\mathbb{E}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k)}^2\right]}^{1/2}}\stackrel{\mathcal{D}}{\to }N(0,1).$$
2. (Eigenvectors) If the unit vector u satisfies that ${\sigma }_{\min }^{-1}{\Vert {\mathbf{v}}_k({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T)\Vert }_{\infty }\to 0$ and $\left|{\mathbf{u}}^T{\mathbf{v}}_k\right|\in [0,1-ϵ]$ for some positive constant, then it holds that

   $$\frac{t_k\left({\mathbf{u}}^T{\widehat{\mathbf{v}}}_k+A_{\mathbf{u},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}^{1/2}\right)-\mathbb{E}\left[\left({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T\right)\mathbf{W}{\mathbf{v}}_k\right]}{{\left\{\mathrm{var}\left[({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k\right]\right\}}^{1/2}}\stackrel{\mathcal{D}}{\to }N(0,1).$$ (21)

   Moreover, it also holds that

   $$\frac{2t_k^2\left({\mathbf{v}}_k^T{\widehat{\mathbf{v}}}_k+A_{{\mathbf{v}}_k,k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}^{1/2}\right)}{{\left[\mathrm{var}({\mathbf{v}}_k^T{\mathbf{W}}^2{\mathbf{v}}_k)\right]}^{1/2}}\stackrel{\mathcal{D}}{\to }N(0,1).$$ (22)

Theorem 2 also gives the asymptotic expansions for the asymptotic mean term $A_{\mathbf{u},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}^{1/2}$. It is seen that if |d_K| diverges to infinity much faster than ${\alpha }_n^2$, then the O(·) terms in the asymptotic expansions of the mean become smaller order terms and thus the following corollary follows immediately from Theorem 3.

Corollary 1.

Assume that Conditions 1–3 hold. For each 1 ≤ k ≤ K, if the unit vector u satisfies that |u^Tv_k| ∈ [0, 1 − ϵ] for some positive constant ϵ and ${\alpha }_n^{-4}{d}_k^2\mathrm{var}[({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k]\to \infty$, then we have

$$\frac{t_k\left({\mathbf{u}}^T{\widehat{\mathbf{v}}}_k-{\mathbf{u}}^T{\mathbf{v}}_k\right)-\mathbb{E}\left[\left({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T\right)\mathbf{W}{\mathbf{v}}_k\right]}{{\left\{\mathrm{var}\left[\left({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T\right)\mathbf{W}{\mathbf{v}}_k\right]\right\}}^{1/2}}\stackrel{\mathcal{D}}{\to }N(0,1).$$ (23)

Moreover, if ${\alpha }_n^{-6}{d}_k^3\mathrm{var}\left({\mathbf{v}}_k^T{\mathbf{W}}^2{\mathbf{v}}_k\right)\to \infty$ then we have

$$\frac{2t_k^2\left({\mathbf{v}}_k^T{\widehat{\mathbf{v}}}_k-1\right)+{\mathbf{v}}_k^T\mathbb{E}{\mathbf{W}}^2{\mathbf{v}}_k}{{\left[\mathrm{var}\left({\mathbf{v}}_k^T{\mathbf{W}}^2{\mathbf{v}}_k\right)\right]}^{1/2}}\stackrel{\mathcal{D}}{\to }N(0,1).$$ (24)

Theorem 3 includes the stochastic block model as a specific case. If X is the affinity matrix from a stochastic block model with K non-overlapping communities and the size of each community is of the same order O(n), then it holds that ‖v_k‖_∞ = O(n^−1/2), d_K = O(n), α_n ≤ n^1/2, and α_n‖v_k‖_∞ ≤ O(1). Thus Condition 3 can be satisfied as long as σ_min ≫ n^−¼, leading to the asymptotic normalities in Theorem 3.

Our Theorem 3 also covers the stochastic block models with overlapping communities. For example, the following network model was considered in Zhang et al. (2015)

$$\mathbb{E}\mathbf{X}=\mathbf{\Theta }\Pi P{\mathbf{\Pi }}^T{\mathbf{\Theta }}^T,$$ (25)

where Θ is an n × n diagonal degree heterogeneity matrix, Π is an n × K community membership matrix, and P is a K × K nonsingular irreducible matrix with unit diagonal entries. Observe that the above model has low-rank mean matrix and thus can be connected to our general form of eigendecomposition $\mathbb{E}\mathbf{X}=\mathbf{H}=\mathbf{V}D{\mathbf{V}}^T$. If the spiked eigenvalues and spiked eigenvectors satisfy that |d_k| = O(n) and ‖v_k‖_∞ = O(n^−1/2) for all 1 ≤ k ≤ K, then Condition 3 can be satisfied when σ_min ≫ n^−¼. Consequently, the asymptotic normalities in Theorem 3 can hold.

3.5. Proofs architecture

The key mathematical tools are from complex analysis and random matrix theory. At a high level, our technical proofs consist of four steps. First, we apply Cauchy’s residue theorem to represent the desired bilinear form ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}$ with 1 ≤ k ≤ K as a complex integral over a contour for a functional of the Green function associated with the original random matrix X = H + W. It is worth mentioning that such an approach was used before to study the asymptotic distributions for linear combinations of eigenvectors in the setting of covariance matrix estimation for the case of i.i.d. Gaussian random matrix coupled with linear dependency. Second, we reduce the problem to one that involves a functional of the new Green function associated with only the noise part W by extracting the spiked part. Such a step enables us to conduct precise high order asymptotic expansions. Third, we conduct delicate high order Taylor expansions for the noise part using new Green function corresponding to the noise part. In this step, we apply the asymptotic expansion directly to the evaluated complex integral over the contour instead of an expansion of the integrand. Such a new way of asymptotic expansion is crucial to our study. Fourth, we bound the variance of ${\mathbf{x}}^T\left({\mathbf{W}}^l-\mathbb{E}{\mathbf{W}}^l\right)\mathbf{y}$ using delicate random matrix techniques. In contrast to just counting the number of certain paths in a graph used in classical random matrix theory literature, we need to carefully bound the individual contributions toward the quantity ${\alpha }_n={\Vert \mathbb{E}{(\mathbf{W}-\mathbb{E}\mathbf{W})}^2\Vert }^{1/2}$; otherwise simple counting leads to rather loose upper bound.

3.6. Comparisons with the statistics literature

In a related work, Tang and Priebe (2018) established the CLT for the entries of eigenvectors of a random adjacency matrix. Our work differs significantly from theirs in at least four important aspects. First, Tang and Priebe (2018) assumed a prior distribution on the mean adjacency matrix, while we assume a deterministic mean matrix. As a result, the asymptotic variance in Tang and Priebe (2018) is determined by the prior distribution and is the same for each entry of an eigenvector, while in our paper the CLT for different entries of an eigenvector can be different and the asymptotic variance depends on all entries of the eigenvector. While Tang and Priebe (2018) also provided the conditional CLT under the setting of the stochastic block model, their result conditions on just one node. Second, our model is much more general than that in Tang and Priebe (2018) in that the spiked eigenvalues can have different orders and different signs. Third, we establish the CLT for the general linear combinations of the components of normalized eigenvectors and the CLT for eigenvalues, while Tang and Priebe (2018) proved the CLT for the rows of ${\mathbf{\Lambda }}^{1/2}{\widehat{\mathbf{V}}}^T$, where $\mathbf{\Lambda }\in {ℝ}^{K\times K}$ is the diagonal matrix formed by K spiked eigenvalues of the adjacency matrix and $\widehat{\mathbf{V}}=\left({\widehat{\mathbf{v}}}_1,\cdots ,{\widehat{\mathbf{v}}}_K\right)$ is the matrix collecting the corresponding eigenvectors of the adjacency matrix. Fourth, through a dedicated analysis of the higher order expansion for the general linear combination ${\mathbf{u}}^T{\widehat{\mathbf{v}}}_k$, we uncover an interesting phase transition phenomenon that the limiting distribution of ${\mathbf{u}}^T{\widehat{\mathbf{v}}}_k$ is different when the deterministic weight vector u is close to or far away from v_k (modulo the sign), which is new to the literature.

Wang and Fan (2017) proved the asymptotic distribution of the linear form ${\mathbf{v}}_i^T{\widehat{\mathbf{v}}}_k$ with 1 ≤ i, k ≤ K, where v_i’s and ${\widehat{\mathbf{v}}}_k$‘s are the spiked population and empirical eigenvectors for some covariance matrix, respectively. Their asymptotic normality results cover the case of ${\mathbf{v}}_1^T{\widehat{\mathbf{v}}}_1$ when K = 1, and ${\mathbf{v}}_i^T{\widehat{\mathbf{v}}}_k$ for 1 ≤ i, k ≤ K with i ≠ k when K ≥ 2. Similarly, Koltchinskii and Lounici (2016) considered the sample covariance matrix under the Gaussian distribution assumption, and derived the asymptotic expansion of the bilinear form ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}$, where x, y are two deterministic unit vectors. They also obtained the asymptotic distribution of ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k$. Different from Wang and Fan (2017) and Koltchinskii and Lounici (2016), in this paper we establish the asymptotic distribution for the general linear combination ${\mathbf{u}}^T{\widehat{\mathbf{v}}}_k$ for the large structured symmetric random matrix from model (1) under fairly weak regularity conditions. Our proof techniques differ from those in Wang and Fan (2017) and Koltchinskii and Lounici (2016), and are also distinct from most of existing ones in the literature.

4. Statistical applications

The new asymptotic expansions and asymptotic distributions of spiked eigenvectors and eigenvalues established in Section 3 have many natural statistical applications. Next we discuss three specific ones. See also Fan et al. (2019) for another application on testing the node membership profiles in network models.

4.1. Detecting the existence of clustering power

One potential application of Theorem 3 is to improve the results on community detection under model setting (25). Spectral methods have been used popularly in the literature for recovering the memberships of nodes in network models. For example, applying the K-means clustering algorithm to the K spiked eigenvectors calculated from the adjacency matrix has been a prevalent method for inferring the memberships of nodes. However, it may not be true that all these K eigenvectors are useful for clustering. For example, if eigenvector ${\mathbf{v}}_k=\mathbf{1}/\sqrt{n}$, then it has zero clustering power and should be dropped in the K-means clustering algorithm. This is especially important in large networks because including a useless high-dimensional eigenvector may significantly increase the noise in clustering. Theorem 3 suggests that we can test the hypothesis $H_0:{\mathbf{v}}_k=\mathbf{1}/\sqrt{n}$ using the test statistic ${\widehat{\mathbf{v}}}_k^T\mathbf{1}$. Then with the aid of Theorem 3, the asymptotic null distribution can be established and the critical value can be calculated. This naturally suggests a method for selecting important eigenvectors in community detection.

4.2. Detecting the existence of denser subgraph

Another application of Theorem 3 is to detect the existence of a denser community in a given random graph, the same problem as studied in Arias-Castro et al. (2014) and Verzelen et al. (2015). Specifically, assume that the data matrix X = (x_ij) is a symmetric adjacency matrix with independent Bernoulli entries on and above the diagonal. Let $\mathbf{H}=\mathbb{E}[\mathbf{X}]$ be the mean adjacency matrix. Consider the following null and alternative hypotheses

$$H_0:\mathbf{H}=p\mathbf{1}{\mathbf{1}}^T\text{vs.}{H}_1:\mathbf{H}=p\mathbf{1}{\mathbf{1}}^T+(q-p)\mathcal{l}{\mathcal{l}}^T,$$

where ℓ is the vector with the first n₁ entries being 1 and all remaining entries being 0, and q ∈ (p, 1]. It can be seen that under the alternative hypothesis, there is a denser subgraph and q measures the connectivity of nodes within it. Arias-Castro et al. (2014) and Verzelen et al. (2015) proposed tests for the above hypothesis in the setting of n₁ = o(n). We focus on the same setting and in addition assume that n⁻¹ ≪ p < q and q ~ p. We next discuss how to exploit our Theorem 3 to test the same hypothesis.

Under the null hypothesis, a natural estimator of p is given by $\widehat{p}=\frac{1}{n(n-1)}{\sum }_{1\le i\le j\le n}{x}_{ij}$. Moreover, direct calculations show that

$${\mathbf{v}}_1^T\mathbb{E}{\mathbf{W}}^2{\mathbf{v}}_1=np(1-p)\text{and}\mathrm{var}\left({\mathbf{v}}_1^T{\mathbf{W}}^2{\mathbf{v}}_1\right)=p(1-p)\left[2(n-1)+p^3+{(1-p)}^3\right].$$ (26)

Thus the mean and variance of ${\mathbf{v}}_1^T{\mathbf{W}}^2{\mathbf{v}}_1$ in (26) can be estimated as

$$n\widehat{p}(1-\widehat{p})\text{and}\widehat{p}(1-\widehat{p})\left[2(n-1)+{\widehat{p}}^3+{(1-\widehat{p})}^3\right],$$ (27)

receptively. In view of (24) in Corollary 1, since v₁ = n^−1/21 under the null hypothesis H₀ : H = p11^T, a natural test statistic for testing H₀ : H = p11^T is given by

$$T_n=\frac{2{\lambda }_1^2(n^{-1/2}{\mathbf{1}}^T{\widehat{\mathbf{v}}}_1-1)+n\widehat{p}(1-\widehat{p})}{{\left[\widehat{p}(1-\widehat{p})\left[2(n-1)+{\widehat{p}}^3+{(1-\widehat{p})}^3\right]\right]}^{1/2}}.$$

It can be seen that since λ₁ ≈ t₁ (see Lemma 3), the asymptotic null distribution of T_n is expected to be N(0, 1) by resorting to (24) in Corollary 1. On the other hand, under the alternative hypothesis, since the leading eigenvector differs from n^−1/21, the term $n^{-1/2}{\mathbf{1}}^T{\widehat{\mathbf{v}}}_1-1$ in the numerator of T_n is expected to take some negative value, and thus T_n is expected to have different asymptotic behavior than N(0, 1). In fact, we provide the proof sketch in Section D.5 of Supplementary Material on the asymptotic null and alternative distributions. In particular, we show that the asymptotic null distribution of T_n is N(0, 1), and if $\frac{n_1^2{(q-p)}^2}{np}+\frac{n_1^2(q-p)}{n}\gg 1$, then T_n → −∞ with asymptotic probability one under the alternative hypothesis.

4.3. Rank inference

Our theory can also be applied to statistical testing on the true rank K of the mean matrix H. Rank inference is an important problem in many high-dimensional network applications. See, for example, Lei (2016), Chen and Lei (2018), and Li et al. (2020), and the importance of the problem discussed therein. Consider the following hypotheses

$$H_0:K=K_0\text{vs.}{H}_1:K>K_0,$$

where K₀ is some prespecified positive integer satisfying K₀ ≤ K. Define

$$\begin{gathered} {\widehat{w}}_{ij}=x_{ij}-\sum _{k=1}^{K_0}{\lambda }_k{\mathbf{e}}_i^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T{\mathbf{e}}_j \\ =w_{ij}-\sum _{k=1}^K\left[{\lambda }_k{\mathbf{e}}_i^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T{\mathbf{e}}_j-d_k{\mathbf{e}}_i^T{\mathbf{v}}_k{\mathbf{v}}_k^T{\mathbf{e}}_j\right]+\sum _{k=K_0+1}^K{\lambda }_k{\mathbf{e}}_i^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T{\mathbf{e}}_j. \end{gathered}$$ (28)

Under the null hypothesis H₀ : K = K₀, the last term in (28) disappears and we can obtain the asympttoic expansion of ${\widehat{w}}_{ij}$ around w_ij explicitly by an application of Theorems 1 and 2. Then under some additional regularity conditions, it is expected that ${\widehat{w}}_{ij}$ is close to w_ij. By the independence of w_ii, i = 1, · · ·, n, it holds that

$$\frac{{\sum }_{i=1}^n{w}_{ii}}{\sqrt{{\sum }_{i=1}^n{w}_{ii}^2}}\stackrel{\mathcal{D}}{\to }N(0,1)\text{as}n\to \infty$$

Since ${\widehat{w}}_{ii}\approx w_{ii}$ under the null hypothesis, the following asymptotic distribution is expected to hold as well

$$T_n:=\frac{{\sum }_{i=1}^n{\widehat{w}}_{ii}}{\sqrt{{\sum }_{i=1}^n{\widehat{w}}_{ii}^2}}\stackrel{\mathcal{D}}{\to }N(0,1).$$ (29)

This naturally suggests a statistical test based on statistic T_n for testing H₀ : K = K₀. Under the alternative hypothesis, since ${\widehat{w}}_{ij}$ contains the smallest K –K₀ spiked eigenvalues and the corresponding eigenvectors, its asymptotic behavior is expected to be different, and consequently, the test can have nontrivial power. In fact, a more sophisticated version of this test constructed based on the off-diagonal entries of $\widehat{\mathbf{W}}$ was investigated recently in Han et al. (2019).

The above asymptotic distribution can also be used to construct confidence intervals for the rank K. To understand this, note that T_n defined in (29) is a function of K₀. Thus an immediate idea for the 100(1–α)% confidence interval construction is to identify all K₀ such that the corresponding T_n falls into the range of [−Φ⁻¹(1–α),Φ^–1(1–α)], where Φ^–1(·) is the inverse distribution function of the standard normal. Similar ideas can also be exploited to construct confidence intervals for other parameters in network models.

5. Simulation studies

In this section, we use simulation studies to verify the validity of our theoretical results. We consider the stochastic block model with K = 2 communities. Assume that the number of nodes is n, the first n/2 nodes belong to the first community, and the rest belong to the second one. Then the adjacency matrix X has the mean structure $\mathbb{E}\mathbf{X}=\mathbf{H}=\mathbf{A}R{\mathbf{A}}^T$, where R is a 2 × 2 matrix of the connectivity probabilities, and $\mathbf{A}=\left({\mathbf{a}}_1,{\mathbf{a}}_2\right)\in {ℝ}^{n\times 2}$ with a₁ = n^−1/2(1^T, 0^T )^T and a₂ = n^−1/2(0^T, 1^T )^T, where $\mathbf{0},\mathbf{1}\in {ℝ}^{n/2}$ are vectors of zeros and ones, respectively. It is worth mentioning that ARA^T is not the eigendecomposition of the mean matrix H, which is why we use different notation than that in model (1).

For the connectivity probability matrix R, we consider the structure

$$\mathbf{R}=r\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right),$$

where parameter r takes 6 different values 0.02, 0.05, 0.1, 0.2, 0.3, and 0.4. A similar model was considered in Abbe et al. (2019) and Lei (2016). For the connectivity matrix X, we simulate its entries on and above the diagonal as independent Bernoulli random variables with means given by the corresponding entries in the mean matrix H, and set the entries below the diagonal to be the same as the corresponding ones above the diagonal. We choose the number of nodes as n = 3000 and repeat the simulations for 10, 000 times.

To verify our theoretical results, for each simulated connectivity matrix X we calculate its eigenvalues and corresponding eigenvectors. For the eigenvalues, we compare the empirical distribution of

$$\frac{{\lambda }_k-t_k}{{\left[\mathrm{var}\left({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\right)\right]}^{1/2}}$$ (30)

with the standard normal distribution, where t_k is the solution to equation (10). The exact expression of $\mathcal{R}\left({\mathbf{v}}_k,{\mathbf{V}}_{-k},z\right){\left[{\left({\mathbf{D}}_{-k}\right)}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{v}}_k,z\right)$ in (10) is complicated. Since this term is much smaller than $\mathcal{R}\left({\mathbf{v}}_k,{\mathbf{v}}_k,z\right)$, we can calculate an approximation of t_k by solving the equation

$$1+d_k\mathcal{R}\left({\mathbf{v}}_k,{\mathbf{v}}_k,z\right)=0$$ (31)

using the Newton–Raphson method. Guided by the theoretical derivations, we use L = 4 in the asymptotic expansion of $\mathcal{R}(\mathbf{x},\mathbf{y},t)$ in (6) for all of our simulation examples. Tables 1–2 summarize the means and standard deviations of (30) with k = 1 and 2 calculated from the 10,000 repetitions as well as the p-values from the Anderson–Darling (AD) test for the normality. Figure 1 presents the histograms of the normalized first and second eigenvalues (i.e., (30) when k = 1 and 2) from the 10,000 repetitions.

Table 1:

Simulation results for $\left({\lambda }_1-t_1\right)/{\left[\mathrm{var}\left({\mathbf{v}}_1^T\mathbf{W}{\mathbf{v}}_1\right)\right]}^{1/2}$

r	0.02	0.05	0.1	0.2	0.3	0.4
Mean	0.0719	0.0149	−0.0068	−0.0080	−0.0024	0.0124
Standard deviation	1.0107	1.0085	0.9927	1.0115	1.0023	1.0125
AD.p-value	0.0725	0.5387	0.6263	0.2342	0.9243	0.2010

Table 2:

Simulation results for $\left({\lambda }_2-t_2\right)/{\left[\mathrm{var}\left({\mathbf{v}}_2^T\mathbf{W}{\mathbf{v}}_2\right)\right]}^{1/2}$

r	0.02	0.05	0.1	0.2	0.3	0.4
Mean	1.0761	0.2552	0.0681	0.0272	0.0093	0.0052
Standard deviation	0.9630	0.9820	0.9872	1.0100	1.0057	1.0005
AD p-value	0.5349	0.6722	0.8406	0.1806	0.0535	0.8341

Figure 1:

Histograms of the normalized eigenvalues (30) when r = 0.4, with the blue curves representing the standard normal density. Left panel: the first eigenvalue; right panel: the second eigenvalue.

For the eigenvectors, we evaluate the asymptotic normality of the linear combination ${\mathbf{u}}^T{\widehat{\mathbf{v}}}_k$ with k = 1 and 2. We experiment with three different values for u: a₁, (1, 0, · · ·, 0)^T, and v_k. When u = a₁ or (1, 0, · · ·, 0)^T, we calculate the normalized statistic

$$\frac{t_k\left({\mathbf{u}}^T{\widehat{\mathbf{v}}}_k+A_{\mathbf{u},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}^{1/2}\right)}{{\left\{\mathrm{var}\left[({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k\right]\right\}}^{1/2}}$$

using the 10,000 simulated data sets, while when u = v_k we calculate the normalized statistic

$$\frac{2t_k^2\left({\mathbf{v}}_k^T{\widehat{\mathbf{v}}}_k+A_{{\mathbf{v}}_k,k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}^{1/2}\right)}{{\left[\mathrm{var}({\mathbf{v}}_k{\mathbf{W}}^2{\mathbf{v}}_k)\right]}^{1/2}}$$

instead. In either of the two cases above, the variance in the denominator is calculated as the sample variance from 2,000 simulated independent copies of the noise matrix W. We compare the empirical distributions of the above two normalized statistics with the standard normal distribution. The simulation results are summarized in Tables 3–8 and Figures 2–3.

Table 3:

Simulation results for ${\mathbf{u}}^T{\widehat{\mathbf{v}}}_1$> with u = a₁

r	0.02	0.05	0.1	0.2	0.3	0.4
Mean	−0.0573	−0.0140	−0.0023	−0.0045	−0.0071	−0.0069
Standard deviation	1.0335	1.0244	1.0011	1.0001	1.0214	1.0016
AD.p-value	0.7879	0.4012	0.2417	0.5300	0.9482	0.9935

Table 8:

Simulation results for ${\mathbf{u}}^T{\widehat{\mathbf{v}}}_2$ with u = (1, 0, ···, 0)^T

r	0.02	0.05	0.1	0.2	0.3	0.4
Mean	0.0622	0.0204	0.0018	−0.0074	−0.0119	−0.0049
Standard deviation	1.1221	1.0537	1.0272	1.0022	1.0088	0.9933
AD p-value	0.0003	0.5853	0.0930	0.6011	0.2423	0.4385

Figure 2:

Histograms corresponding to the first eigenvector ${\widehat{\mathbf{v}}}_1$ when r = 0.4, with the blue curves representing the standard normal density. Left panel:${\mathbf{u}}_1^T{\widehat{\mathbf{v}}}_2$; middle panel:${\mathbf{v}}_2^T{\widehat{\mathbf{v}}}_2$; right panel:${\mathbf{u}}_3^T{\widehat{\mathbf{v}}}_1$, where u₁ = a₁ and u₃ = (1,0, · · ·, 0)^T.

Figure 3:

Histograms corresponding to the second eigenvector ${\widehat{\mathbf{v}}}_2$ when r = 0.4, with the blue curves representing the standard normal density. Left panel:${\mathbf{u}}_1^T{\widehat{\mathbf{v}}}_2$; middle panel:${\mathbf{v}}_2^T{\widehat{\mathbf{v}}}_2$; right panel:${\mathbf{u}}_3^T{\widehat{\mathbf{v}}}_1$, where u₁ = a₁ and u₃ = (1,0, · · ·, 0)^T.

Our simulation results in Figure 1 and Tables 1–2 suggest that the normalized spiked eigenvalues have distributions very close to standard Gaussian which supports our results in Theorem 1. Indeed, such a large p-value is extremely impressive given the “sample size” (the number of simulations is 10,000). In general, the simulation results for the eigenvectors support our theoretical findings in Section 3. However, the results corresponding to the first spiked eigenvector ${\widehat{v}}_1$ (Tables 3–5) are better than those for the second spiked eigenvector ${\widehat{v}}_2$ (Tables 4–8). This is reasonable since for the larger spiked eigenvalue, the negligible terms that we dropped in the proofs of the asymptotic normality become relatively smaller and thus have smaller finite-sample effects on the asymptotic distributions. For the linear form ${\mathbf{u}}^T{\widehat{\mathbf{v}}}_k$, when u = v_k the convergence to standard normal is slower when compared to the case of u ≠ v_k. This again supports our theoretical findings in Section 3 and explains why we need to separate the cases of u = v_k and u ≠ v_k. Such effect is especially prominent for ${\mathbf{v}}_2^T{\widehat{\mathbf{v}}}_2$, whose sample mean is −11.8020 when r = 0.02 as shown in Table 7. However, it is seen from the same table (and other tables) that as the spiked eigenvalue increases with r, the distribution gets closer and closer to standard Gaussian.

Table 5:

Simulation results for ${\mathbf{u}}^T{\widehat{\mathbf{v}}}_1$ with u = (1, 0, ···, 0)^T

r	0.02	0.05	0.1	0.2	0.3	0.4
Mean	0.0025	0.0021	0.0003	0.0105	0.0061	−0.0122
Standard deviation	1.0432	1.0354	0.9871	1.0016	1.0205	0.9898
AD.p-value	0.0044	0.4877	0.3752	0.1514	0.1304	0.3400

Table 4:

Simulation results for ${\mathbf{v}}_1^T{\widehat{\mathbf{v}}}_1$

r	0.02	0.05	0.1	0.2	0.3	0.4
Mean	−1.3288	−0.4817	−0.1900	−0.0742	−0.0409	−0.0186
Standard deviation	1.0940	1.0545	1.0338	0.9749	1.0030	1.0005
AD.p-value	0.0582	0.4251	0.0251	0.0225	0.3312	0.2912

Table 7:

Simulation results for ${\mathbf{v}}_2^T{\widehat{\mathbf{v}}}_2$

r	0.02	0.05	0.1	0.2	0.3	0.4
Mean	−11.8020	−4.3274	−2.0057	−0.7447	−0.3526	−0.1650
Standard deviation	1.3775	1.1192	1.0980	1.0343	1.0104	1.0089
AD p-value	0.0000	0.0011	0.0422	0.3964	0.4980	0.1186

6. A more general asymptotic theory

As mentioned before, the asymptotic theory on the spiked eigenvectors in terms of the general linear combination and on the spiked eigenvalues presented in Section 3 is in fact a consequence of a more general asymptotic theory on the spiked eigenvectors in terms of the bilinear form. In this section, we focus our attention on such a more general asymptotic theory for the bilinear form ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}$ with 1 ≤ k ≤ K, where x and y are two arbitrary n-dimensional unit vectors. See Sections 3.5 and 3.6 for detailed discussions on the technical innovations of our novel ATE theoretical framework and comparisons with the existing literature on the asymptotic distributions of eigenvectors.

For technical reasons, we will break our main results on the asymptotic distributions of the bilinear form ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}$ down to two theorems, where we consider in Theorem 4 the case when either vector x or vector y is sufficiently further away from the population eigenvector v_k, and then we study in Theorem 5 the case when both vectors x and y are very close to v_k The technical treatments for these two cases are different since in the latter scenario, the first order term which determines the asymptotic distribution in Theorem 4 vanishes, and thus we need to consider higher order expansions to obtain the asymptotic distribution in Theorem 5. Let ${\mathbf{J}}_{\mathbf{x},\mathbf{y},k,t_k}$, ${\mathbf{L}}_{\mathbf{x},\mathbf{y},k,t_k}$, and ${\mathbf{Q}}_{\mathbf{X},\mathbf{y},k,t_k}$ be the three rank one matrices given in (113)–(115), respectively, in the proof of Theorem 5 in Section A.6. Denote by ${\sigma }_k^2=\mathrm{var}\left[\mathrm{tr}\left(\mathbf{W}{\mathbf{J}}_{\mathbf{x},\mathbf{y},k,t_k}\right)\right]$ and

$${\tilde{\sigma }}_k^2=\mathrm{var}\left\{\mathrm{tr}\left[\mathbf{W}{\mathbf{J}}_{\mathbf{x},\mathbf{y},k,t_k}-\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{L}}_{\mathbf{x},\mathbf{y},k,t_k}\right]+\mathrm{tr}(\mathbf{W}{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathrm{tr}\left(\mathbf{W}{\mathbf{Q}}_{\mathbf{x},\mathbf{y},k,t_k}\right)\right\}.$$ (32)

Both of the quantities above play an important role in our more general asymptotic theory.

Theorem 4.

Assume that Conditions 1–2 hold and x and y are two n-dimensional unit vectors. Then for each 1 ≤ k ≤ K, if ${\sigma }_k^2\gg t_k^{-4}{\alpha }_n^2{\left(\left|A_{\mathbf{x},k,t_k}\right|+\left|A_{\mathbf{y},k,t_k}\right|\right)}^2+t_k^{-4}$ we have the asymptotic expansion

$${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}=a_k+\mathrm{tr}\left(\mathbf{W}{\mathbf{J}}_{\mathbf{x},\mathbf{y},k,t_k}\right)+O_p\left\{t_k^{-2}{\alpha }_n\left(\left|A_{\mathbf{x},k,t_k}\right|+\left|A_{\mathbf{y},k,t_k}\right|\right)+t_k^{-2}\right\},$$ (33)

where the quantity $a_k=A_{\mathbf{x},k,t_k}{A}_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}$.

The assumption of ${\sigma }_k^2\gg t_k^{-4}{\alpha }_n^2{\left(\left|A_{\mathbf{x},k,t_k}\right|+\left|A_{\mathbf{y},k,t_k}\right|\right)}^2+t_k^{-4}$ in Theorem 4 requires the variance of random variable $\mathrm{tr}\left(\mathbf{W}{\mathbf{J}}_{\mathbf{x},\mathbf{y},k,t_k}\right)$ not too small, which at high level, requires that either vector x or y is sufficiently faraway from the population eigenvector v_k. If σ_ij ~ 1 for each (i, j) pair, then such an assumption restricts essentially that $\Vert {\mathbf{J}}_{\mathbf{X},\mathbf{y},k,t_k}\Vert$ should not be too close to zero. This in turn ensures that the first order expansion is sufficient for deriving the asymptotic normality of ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}$. Theorem 4 also entails that a simple upper bound for ${\tilde{\sigma }}_k$ as defined in (32) can be shown to be $O\left(t_k^{-2}{\alpha }_n\right)$.

Theorem 5.

Assume that Conditions 1–2 hold and x and y are two n-dimensional unit vectors. Then for each 1 ≤ k ≤ K, if ${\sigma }_k^2=O\left({\tilde{\sigma }}_k^2\right)$ and ${\tilde{\sigma }}_k^2\gg t_k^{-6}{\alpha }_n^4{\left(\left|A_{\mathbf{x},k,t_k}\right|+\left|A_{\mathbf{y},k,t_k}\right|\right)}^2+t_k^{-6}$ we have the asymptotic expansion

$$\begin{gathered} {\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}=a_k+\mathrm{tr}\left[\mathbf{W}{\mathbf{J}}_{\mathbf{x},\mathbf{y},k,t_k}-\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{L}}_{\mathbf{x},\mathbf{y},k,t_k}\right]+\mathrm{tr}(\mathbf{W}{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathrm{tr}\left(\mathbf{W}{\mathbf{Q}}_{\mathbf{x},\mathbf{y},k,t_k}\right) \\ +O_p\left\{{\left|t_k\right|}^{-3}{\alpha }_n^2\left(\left|A_{\mathbf{x},k,t_k}\right|+\left|A_{\mathbf{y},k,t_k}\right|\right)+{\left|t_k\right|}^{-3}\right\}, \end{gathered}$$ (34)

where the quantity a_k is given in (33).

The ATE theoretical framework for the more general asymptotic theory established in Theorems 4 and 5 is empowered by the following two technical lemmas.

Lemma 4.

For any n-dimensional unit vectors x and y, we have

$${\mathbf{x}}^T\left({\mathbf{W}}^l-\mathbb{E}{\mathbf{W}}^l\right)\mathbf{y}=O_p\left(\min \left\{{\alpha }_n^{l-1},d_{\mathbf{X}}{\alpha }_n^l,d_{\mathbf{y}}{\alpha }_n^l\right\}\right)$$ (35)

with l ≥ 1 some bounded positive integer and dx = ‖x‖_∞.

Lemma 5.

For any n-dimensional unit vectors x and y, we have $\mathbb{E}{\mathbf{x}}^T{\mathbf{W}}^l\mathbf{y}=O(1)$ and

$$\mathbb{E}{\mathbf{x}}^T{\mathbf{W}}^l\mathbf{y}=O\left({\alpha }_n^l\right)$$ (36)

with l ≥ 2 some bounded positive integer.

The detailed proofs of Lemmas 4 and 5 are provided in Sections B.5 and B.6 of Supplementary Material. Our delicate technical arguments therein establish useful refinements to the classical idea of counting the number of nonzero terms from the random matrix theory. In particular, Lemma 4 is the key building block for high order Taylor expansions that involve polynomials of quantities in the lemma with different choices of (x, y, l).

7. Discussions

In contrast to the immense literature on the asymptotic distributions for eigenvalues of large spiked random matrices, the counterpart asymptotic theory for eigenvectors has remained largely underdeveloped in statistics literature for years. Yet such a theory is much desired for understanding the precise asymptotic properties of various statistical and machine learning algorithms that build upon the spectral information of the eigenspace constructed from observed data matrix. Our work in this paper provides a first attempt with a general ATE theoretical framework for underpinning the precise asymptotic expansions and asymptotic distributions for spiked eigenvectors and spiked eigenvalues of large spiked random matrices with diverging spikes. Our results complement existing ones in the RMT literature as well as the networks literature.

The family of models in our ATE framework includes many popularly used ones for large-scale applications including network analysis and text analysis such as the stochastic block models with or without overlapping communities and the topic models. Our general asymptotic theory for eigenvectors can be exploited to develop new useful tools for precise statistical inference in these applications. It would be interesting to investigate the problem of reproducible large-scale inference as in Barber and Candès (2015); Candès et al. (2018); Lu et al. (2018); Fan et al. (2019); Fan et al. (2019) in these model settings. It would also be interesting to develop a general method to determine the rank and provide robust rank inference in such high-dimensional low-rank models. These extensions are beyond the scope of the current paper and will be interesting topics for future research.

Table 6:

Simulation results for ${\mathbf{u}}^T{\widehat{\mathbf{v}}}_2$ with u = a₁

r	0.02	0.05	0.1	0.2	0.3	0.4
Mean	4.2611	1.0129	0.3067	0.0745	0.0219	0.0037
Standard deviation	1.2384	1.0952	1.0294	1.0098	1.0280	1.0044
AD p-value	0.3829	0.7535	0.3759	0.4105	0.9129	0.9873

A. Proofs of main results

Recall that Condition 2 involves two scenarios of the spike strength. We will first prove all the results under scenario i). Then in Section D of Supplementary Material, we will adapt the proofs to show that the same results also hold under scenario ii). We provide the proofs of Theorems 1–5 and Corollary 1 in this appendix. Additional technical details including the proofs of all the lemmas and further discussions on when the asymptotic normality can hold for the asymptotic expansion in Theorem 5 are contained in the Supplementary Material.

A.1. Proof of Theorem 1

The results on the asymptotic distributions of spiked eigenvalues in Theorem 1 are in fact a consequence of those on the asymptotic expansions and asymptotic distributions for the spiked eigenvectors, where a more general asymptotic theory of the eigenvectors is presented in Theorems 4–5 in Section 6. Let us define a matrix-valued function that is referred to as the Green function associated with only the noise part W

$$\mathbf{G}(z)={(\mathbf{W}-z\mathbf{I})}^{-1}$$ (37)

for z in the complex plane $ℂ$, where I stands for the identity matrix of size n. Recall that λ₁, · · ·, λ_n are the eigenvalues of matrix X and ${\widehat{\mathbf{v}}}_1,\cdots ,{\widehat{\mathbf{v}}}_n$ are the corresponding eigenvectors. By Weyl’s inequality, it holds that max|λ_i – d_i| ≤ ‖W‖. Thus, in view of Condition 2 and Lemma 6 in Supplementary Material, all the spiked eigenvalues λ_k with 1 ≤ k ≤ K of the observed random matrix X have magnitudes of larger order than the eigenvalues of the noise matrix W with significant probability as the matrix size n increases. This entails that with significant probability, matrices G(λ_k) with 1 ≤ k ≤ K are well defined and nonsingular. For the rest of this proof, we restrict all the derivations on such an event that holds with asymptotic probability one.

It follows from the definition of the eigenvalue, the representation X = H+W = VDV^T + W, (37), and the properties of the determinant function det(·) that for each 1 ≤ k ≤ K,

$$\begin{gathered} 0=\det \left(\mathbf{X}-{\lambda }_k\mathbf{I}\right)=\det \left(\mathbf{W}-{\lambda }_k\mathbf{I}+\mathbf{V}\mathbf{D}{\mathbf{V}}^T\right)=\det \left[{\mathbf{G}}^{-1}\left({\lambda }_k\right)+\mathbf{V}\mathbf{D}{\mathbf{V}}^T\right] \\ =\det \left[{\mathbf{G}}^{-1}\left({\lambda }_k\right)\right]\det \left[\mathbf{I}+\mathbf{G}\left({\lambda }_k\right)\mathbf{V}\mathbf{D}{\mathbf{V}}^T\right], \end{gathered}$$

which leads to det[I + G(λ_k)VDV^T ] = 0 since det[G⁻¹(λ_k)] = det[G(λ_k)]⁻¹ is nonzero. Using the identity det(I + AB) = det(I + BA) for matrices A and B, we obtain for each 1 ≤ k ≤ K,

$$0=\det \left[\mathbf{I}+\mathbf{G}\left({\lambda }_k\right)\mathbf{V}\mathbf{D}{\mathbf{V}}^T\right]=\det \left[\mathbf{I}+\mathbf{D}{\mathbf{V}}^T\mathbf{G}\left({\lambda }_k\right)\mathbf{V}\right],$$ (38)

where the second I represents an identity matrix of size K and we slightly abuse the notation for simplicity. Since the diagonal matrix D is nonsingular by assumption, it follows from (38) that

$$\det \left[d_k{\mathbf{V}}^T\mathbf{G}\left({\lambda }_k\right)\mathbf{V}+d_k{\mathbf{D}}^{-1}\right]=d_k\det \left({\mathbf{D}}^{-1}\right)\det \left[\mathbf{I}+\mathbf{D}{\mathbf{V}}^T\mathbf{G}\left({\lambda }_k\right)\mathbf{V}\right]=0$$ (39)

for each 1 ≤ k ≤ K.

By the asymptotic expansions in (79), Lemmas 4 and 5, and Weyl’s inequality max|λ_k – d_k| ≤ ‖W‖, we have for j ≠ ℓ, $d_k{\mathbf{v}}_j^T\mathbf{G}\left({\lambda }_k\right){\mathbf{v}}_{\mathcal{l}}=-d_k{O}_p\left({\lambda }_k^{-2}\right)=O_p\left(1/\left|d_k\right|\right)$. Thus, we can see that all off diagonal entries of matrix d_kV^T G(λ_k)V+d_kD⁻¹ in (39) are of order O_p(1/|d_k|). For j ≠ k, the jth diagonal entry of d_kV^T G(λ_k)V + d_kD⁻¹ equals $d_k{\mathbf{v}}_j^T\mathbf{G}\left({\lambda }_k\right){\mathbf{v}}_j+d_k/d_j$. By (78) and Lemma 4, we have $d_k{\mathbf{v}}_j^T\mathbf{G}\left({\lambda }_k\right){\mathbf{v}}_j+1=o_p(1)$. Moreover, by Condition 2 |d_k/d_j – 1| ≥ c for some positive constant c. Hence, all these diagonal entries but the kth one are of order at least O_p(1). Thus the matrix ${\left(d_k{\mathbf{v}}_i^T\mathbf{G}\left({\lambda }_k\right){\mathbf{v}}_j+{\delta }_{ij}{d}_k/d_i\right)}_{1\le i,j\le K,i,j\ne k}$ is invertible with significant probability, where δ_ij = 1 when i = j and 0 otherwise. Recall the determinant identity for block matrices from linear algebra

$$\det \left(\begin{array}{cc}{\mathbf{A}}_{11}& {\mathbf{A}}_{12}\\ {\mathbf{A}}_{21}& {\mathbf{A}}_{22}\end{array}\right)=\det \left({\mathbf{A}}_{22}\right)\det \left({\mathbf{A}}_{11}-{\mathbf{A}}_{12}{\mathbf{A}}_{22}^{-1}{\mathbf{A}}_{21}\right)$$

when the lower right block matrix A₂₂ is nonsingular. Treating the kth diagonal entry of d_kV^T G(λ_k)V + d_kD⁻¹ as the first block, we have with significant probability

$$\det \left[d_k{\mathbf{V}}^T\mathbf{G}\left({\lambda }_k\right)\mathbf{V}+d_k{\mathbf{D}}^{-1}\right]=0$$ (40)

entailing $d_k{\mathbf{v}}_k^T\mathbf{G}\left({\lambda }_k\right){\mathbf{v}}_k+1=d_k{\mathbf{v}}_k^T{\mathbf{F}}_k\left({\lambda }_k\right){\mathbf{v}}_k$, where ${\mathbf{F}}_k(z)=\mathbf{G}(z){\mathbf{V}}_{-k}{\left[{\mathbf{D}}_{-k}^{-1}+{\mathbf{V}}_{-k}^T\mathbf{G}(z){\mathbf{V}}_{-k}\right]}^{-1}\cdot {\mathbf{V}}_{-k}^T\mathbf{G}(z)$ and A_−k denotes the submatrix of matrix A by removing the kth column. In light of (40) and the solution ${\widehat{t}}_k$ to equation (94) in the proof of Theorem 4 in Section A.5, it holds from the uniqueness of ${\widehat{t}}_k$ that

$${\lambda }_k={\widehat{t}}_k.$$ (41)

Therefore, combining equality (41) with asymptotic expansion ${\widehat{t}}_k-t_k={\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n/t_k\right)$ obtained in (99) completes the proof of Theorem 1.

A.2. Proof of Theorem 2

The results on the asymptotic distributions of spiked eigenvectors in Theorem 2 are also an implication of a more general asymptotic theory of the eigenvectors presented in Theorems 4–5 in Section 6 on the delicate asymptotic expansions and asymptotic distributions for the spiked eigenvectors. Recall that $\widehat{\mathbf{V}}=\left({\widehat{\mathbf{v}}}_1,\cdots ,{\widehat{\mathbf{v}}}_K\right)$ with ${\widehat{\mathbf{v}}}_k$ for 1 ≤ k ≤ K the empirical spiked eigenvectors of the observed random matrix X. Without loss of generality, let us choose the direction of eigenvector ${\widehat{\mathbf{v}}}_k$ such that ${\widehat{\mathbf{v}}}_k^T{\mathbf{v}}_k\ge 0$. Clearly, fixing the direction of ${\widehat{\mathbf{v}}}_k$ does not affect the distribution of ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}$; that is, its distribution stays the same when $-{\widehat{\mathbf{v}}}_k$ is chosen as the eigenvector. We will separately consider the two cases of ${\mathbf{v}}_k^T{\widehat{\mathbf{v}}}_k$ and ${\mathbf{u}}^T{\widehat{\mathbf{v}}}_k$ with u ≠ v_k, where the former relies on the second order expansion given in (111) in the proof of Theorem 5 in Section A.6, and the latter utilizes the first order expansion given in (107) in the proof of Theorem 4 in Section A.5.

We first consider ${\mathbf{v}}_k^T{\widehat{\mathbf{v}}}_k$. Choosing x = y = v_k in Theorem 4 gives $a_k=A_{{\mathbf{v}}_k,k,t_k}^2{\tilde{\mathcal{P}}}_{k,t_k}$. By Lemma 5, it holds that

$$\mathcal{P}\left({\mathbf{v}}_k,{\mathbf{v}}_k,t_k\right)=-\sum _{l=0,l\ne 2}^L\frac{{\mathbf{v}}_k^T\mathbb{E}{\mathbf{W}}^l{\mathbf{v}}_k}{t_k^l}=-1+O\left({\alpha }_n^2/t_k^2\right)$$ (42)

and

$$\Vert \mathcal{P}\left({\mathbf{v}}_k,{\mathbf{V}}_{-k},t_k\right)\Vert =\Vert -\sum _{l=0,l\ne 2}^L\frac{{\mathbf{v}}_k^T\mathbb{E}{\mathbf{W}}^l{\mathbf{V}}_{-k}}{t_k^l}\Vert =O\left({\alpha }_n^2/t_k^2\right).$$ (43)

Moreover, recalling the definition of $A_{\mathbf{u},k,t_k}$ in (9), $A_{\mathbf{u},k,t_k}$ can be rewritten as

$$A_{{\mathbf{V}}_k,k,t_k}=\mathcal{P}\left({\mathbf{v}}_k,{\mathbf{v}}_k,t\right)-t_k^{-1}\mathcal{P}\left({\mathbf{v}}_k,{\mathbf{V}}_{-k},t_k\right){\left({\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right)}^{-1}\mathcal{P}\left({\mathbf{V}}_{-k},{\mathbf{v}}_k,t_k\right).$$

Therefore, by (42)–(43), (A.16), and (91), we have

$$A{\mathbf{v}}_k,k,t_k=-1+O\left({\alpha }_n^2/t_k^2\right)\text{and}{\tilde{\mathcal{P}}}_{k,t_k}=1+O\left({\alpha }_n^2/t_k^2\right).$$ (44)

Now recall the second order expansion of ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k\mathbf{y}$ given in (111) in the proof of Theorem 5. We next calculate the orders of each term in the expansion (111). First, we consider ${\mathbf{b}}_{{\mathbf{v}}_k,k,t_k}^T$. By (43), (A.16), and the definition (7), we have

$$\Vert {\mathbf{b}}_{{\mathbf{v}}_k,k,t_k}^T-{\mathbf{v}}_k^T\Vert =\Vert \mathcal{R}\left({\mathbf{v}}_k,{\mathbf{V}}_{-k},t_k\right){\left({\left({\mathbf{D}}_{-k}\right)}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right)}^{-1}{\mathbf{V}}_{-k}^T\Vert =O\left({\alpha }_n^2/t_k^2\right).$$ (45)

This together with (44) entails that

$$\Vert {\mathbf{b}}_{{\mathbf{v}}_k,k,t_k}^T+A_{{\mathbf{v}}_k,k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T\Vert =O\left({\alpha }_n^2/t_k^2\right).$$ (46)

It follows from Lemma 4 and (44) that

$$A_{{\mathbf{v}}_k,k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}({\mathbf{b}}_{{\mathbf{v}}_k,k,t_k}^T+A_{{\mathbf{v}}_k,k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k/t_k=O_p\left({\alpha }_n^2/{\left|t_k\right|}^3\right),$$

$$\begin{array}{l}{\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-2}\left[2{\tilde{\mathcal{P}}}_{k,t_k}\left(A_{{\mathbf{v}}_k,k,t_k}{\mathbf{b}}_{{\mathbf{v}}_k,k,t_k}^T+A_{{\mathbf{v}}_k,k,t_k}{\mathbf{b}}_{{\mathbf{v}}_k,k,t_k}^T\right)\mathbf{W}{\mathbf{v}}_k{\mathbf{v}}_k^T+{\mathbf{b}}_{{\mathbf{v}}_k,k,t_k}^T\mathbf{W}{\mathbf{v}}_k{\mathbf{b}}_{{\mathbf{v}}_k,k,t_k}^T\right]\mathbf{W}{\mathbf{v}}_k\hfill \\ +2A_{{\mathbf{v}}_k,k,t_k}{A}_{\mathbf{y},k,t_k}{t}_k^{-2}{\left({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\right)}^2\hfill \\ +A_{{\mathbf{v}}_k,k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}\{t_k^{-2}{\mathbf{x}}^T\mathbf{W}{\mathbf{v}}_k{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k-t_k^{-2}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\mathcal{R}\left({\mathbf{v}}_k,{\mathbf{V}}_{-k},t\right)\hfill \\ \times {\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\mathbf{W}{\mathbf{v}}_k\}\hfill \\ +A_{{\mathbf{v}}_k,k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}\{t_k^{-2}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k-t_k^{-2}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\mathcal{R}\left({\mathbf{v}}_k,{\mathbf{V}}_{-k},t\right)\hfill \\ \times {\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\mathbf{W}{\mathbf{v}}_k\}\hfill \\ +A_{{\mathbf{v}}_k,k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-2}\mathcal{R}\left({\mathbf{v}}_k,{\mathbf{V}}_{-k},t_k\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k\hfill \\ +A_{{\mathbf{v}}_k,t,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-2}\mathcal{R}\left({\mathbf{v}}_k,{\mathbf{V}}_{-k},t_k\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k\hfill \\ =O_p\left(\frac{1}{{\left|t_k\right|}^2}+\frac{{\alpha }_n}{{\left|t_k\right|}^3}\right),\hfill \end{array}$$

and

$$\begin{gathered} {\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-2}\left(A_{{\mathbf{v}}_k,k,t_k}{\mathbf{x}}^T+A_{{\mathbf{v}}_k,k,t_k}{\mathbf{y}}^T\right)\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k+3t_k^{-2}{A}_{{\mathbf{v}}_k,k,t_k}{A}_{{\mathbf{v}}_k,k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k \\ =\frac{{\mathbf{v}}_k^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k}{t_k^2}+O_p\left({\alpha }_n^3/t_k^4\right). \end{gathered}$$

Substituting the above equations into (111) results in

$${\mathbf{v}}_k^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T{\mathbf{v}}_k-A_{{\mathbf{v}}_k,k,t_k}^2{\tilde{\mathcal{P}}}_{k,t_k}=-{\mathbf{v}}_k^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k/t_k^2+O_p\left({\left|t_k\right|}^{-2}+{\alpha }_n^2/{\left|t_k\right|}^3\right),$$ (47)

where the leading term of the asymptotic expansion now depends on the second moments of the noise matrix W. Recall that ${\mathbf{v}}_k^T{\widehat{\mathbf{v}}}_k\ge 0$. By (44) and (47) we have

$$\begin{gathered} {\mathbf{v}}_k^T{\widehat{\mathbf{v}}}_k+A_{{\mathbf{v}}_k,k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}^{1/2}=-\frac{{\mathbf{v}}_k^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k}{2A_{{\mathbf{v}}_k,k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}^{1/2}{t}_k^2}+O_p\left({\alpha }_n^2/t_k^3\right) \\ =-\frac{{\mathbf{v}}_k^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k}{2t_k^2}+O_p\left({\left|t_k\right|}^{-2}+{\alpha }_n^2/{\left|t_k\right|}^3\right). \end{gathered}$$ (48)

We now consider an arbitrary unit vector $\mathbf{u}\in {ℝ}^n$ with |u^T v_k| ∈ [0, 1) for investigating the asymptotic distributions of the general linear combinations ${\mathbf{u}}^T{\widehat{\mathbf{v}}}_k$. Recall the first order expansion given in (107) in the proof of Theorem 4 and (46) that

$$\begin{gathered} {\mathbf{u}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T{\mathbf{v}}_k-A_{\mathbf{u},k,t_k}{A}_{{\mathbf{v}}_k,k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}=-A_{{\mathbf{v}}_k,k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}({\mathbf{b}}_{\mathbf{u},k,t_k}^T+A_{\mathbf{u},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k/t_k \\ +O_p\left({\alpha }_n/t_k^2\right). \end{gathered}$$ (49)

Then dividing (49) by ${\mathbf{v}}_k^T{\widehat{\mathbf{v}}}_k$ and using (44) and (48), we can deduce that

$$\begin{gathered} {\mathbf{u}}^T{\widehat{\mathbf{v}}}_k+A_{\mathbf{u},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}^{1/2}={\tilde{\mathcal{P}}}_{k,t_k}^{1/2}({\mathbf{b}}_{\mathbf{u},k,t_k}^T+A_{\mathbf{u},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k/t_k+O_p\left({\alpha }_n/t_k^2\right) \\ =({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k/t_k+O_p\left({\alpha }_n/t_k^2\right). \end{gathered}$$ (50)

In view of the asymptotic expansions in (50) and (48), we can see that the desired asymptotic normalities in the two parts of Theorem 2 follow from the conditions of Lemmas 1 or 2. More specifically, for (50) if ${\alpha }_n^{-2}{d}_k^2\mathrm{var}\left[({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k\right]\to \infty$, then we have ${\alpha }_n/t_k^2\ll {\left\{\mathrm{var}\left[({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k\right]\right\}}^{1/2}$ and thus the first part of Theorem 2 in (17) holds in view of (50). Furthermore, if $({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k$ is W¹-CLT, then $({\mathbf{b}}_{\mathbf{u},k,t_k}-{\mathbf{v}}_k{\mathbf{v}}_k^T\mathbf{u},{\mathbf{v}}_k)$ is also W¹-CLT and thus we have

$$\frac{t_k\left({\mathbf{u}}^T{\widehat{\mathbf{v}}}_k+A_{\mathbf{u},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}^{1/2}\right)-\mathbb{E}\left[({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k\right]}{{\left\{\mathrm{var}\left[({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k\right]\right\}}^{1/2}}\stackrel{\mathcal{D}}{\to }N(0,1).$$

Similarly, the second part of Theorem 2 in (18) also holds under the condition $({\alpha }_n^{-4}{d}_k^2+\text{1})\mathrm{var}\left[{\mathbf{v}}_k^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k\right]\to \infty$ and the CLT holds if (v_k, v_k) is W²-CLT. This concludes the proof of Theorem 2.

A.3. Proof of Theorem 3

The results on the asymptotic distributions of spiked eigenvalues and spiked eigenvectors in Theorem 3 are an application of those in Theorems 1 and 2 for a more specific structure of the low rank model (1), including the stochastic block model with both non-overlapping and overlapping communities as special cases.

First, note that (15) implies that the condition of Lemma 1 holds for ${\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k$ under Condition 3. Consequently, (v_k, v_k) is W¹-CLT. In addition, (15) ensures that $\mathbb{E}({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k-{\mathbb{E}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k)}^2\gg {\alpha }_n^2/d_k^2$ under Condition 3. Therefore, it follows from Theorem 1 that the first result of Theorem 3 holds. Recall that in (A.12), s_x,y is defined as the expected value of the conditional variance of ${\mathbf{v}}_k^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k$. By definition, we have $\mathrm{var}\left[{\mathbf{v}}_k^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k\right]\ge s_{\mathbf{x},\mathbf{y}}\ge c{\sigma }_{\min }^{2}n$. Thus the condition $\left({\alpha }_n^{-4}{d}_k^2+1\right)\mathrm{var}\left[{\mathbf{v}}_k^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k\right]\to \infty$ in Theorem 2 is ensured by the assumptions

$${\sigma }_{\min }^{2}n\to \infty ,\frac{\left|d_K\right|{\sigma }_{\min }}{{\alpha }_n}\to \infty ,{\alpha }_n\le n^{1/2}$$

in Condition 3. Moreover, by (A.13) we can see that the conditions of Lemma 2 are satisfied for ${\mathbf{v}}_k^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k$ under Condition 3. Thus (v_k, v_k) is W²-CLT. Therefore, (22) holds by an application of (18) in Theorem 2.

It remains to show that the condition

$$\mathrm{var}\left[({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k\right]\gg {\alpha }_n^2/d_k^2$$ (51)

in Theorem 2 can be guaranteed by Condition 3. Then the expansion in (17) holds. Moreover, the condition ${\sigma }_{\min }^{-1}{\Vert {\mathbf{v}}_k\left[{\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T\right]\Vert }_{\infty }\to 0$ ensures that $\left({\mathbf{b}}_{\mathbf{u},k,t_k}-{\mathbf{v}}_k{\mathbf{v}}_k^T\mathbf{u},{\mathbf{v}}_k\right)$ is W¹-CLT. Combining these results entails that the asymptotic normality (21) holds. Now we proceed to verify (51). Consider an arbitrary unit vector $\mathbf{u}\in {ℝ}^n$ satisfying $\left|{\mathbf{u}}^T{\mathbf{v}}_k\right|\in [0,1-ϵ]$ for some positive constant ϵ. Recalling the definition of b_u,k,t in (8), we have ${\mathbf{b}}_{\mathbf{u},k,t}^T{\mathbf{v}}_k=\{{\mathbf{u}}^T-\mathcal{R}\left(\mathbf{u},{\mathbf{V}}_{-k},t\right){\left[{\left({\mathbf{D}}_{-k}\right)}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\}{\mathbf{v}}_k={\mathbf{u}}^T{\mathbf{v}}_k$. Thus it holds that ${\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T={\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{b}}_{\mathbf{u},k,t_k}^T{\mathbf{v}}_k{\mathbf{v}}_k^T={\mathbf{b}}_{\mathbf{u},k,t_k}^T(\mathbf{I}-{\mathbf{v}}_k{\mathbf{v}}_k^T)$. Moreover, similar to (15) we can show that

$${\left[\mathbb{E}{\left({\mathbf{u}}^T\mathbf{W}{\mathbf{v}}_k-\mathbb{E}{\mathbf{u}}^T\mathbf{W}{\mathbf{v}}_k\right)}^2\right]}^{1/2}\ge {\sigma }_{\min }{\left(2-2{\Vert {\mathbf{v}}_k\Vert }_{\infty }^2\right)}^{1/2}.$$ (52)

This ensures that there exists some positive constant c₁ such that

$$\begin{gathered} \mathrm{var}{\left[({\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k\right]}^2\ge {\sigma }_{{\min }^2}\left(2-2{\Vert {\mathbf{v}}_k\Vert }_{\infty }^2\right){\Vert {\mathbf{b}}_{\mathbf{u},k,t_k}^T-{\mathbf{u}}^T{\mathbf{v}}_k{\mathbf{v}}_k^T\Vert }^2 \\ \ge {\sigma }_{{\min }^2}\left(2-2{\Vert {\mathbf{v}}_k\Vert }_{\infty }^2\right){\Vert {\mathbf{b}}_{\mathbf{u},k,t_k}^T(\mathbf{I}-{\mathbf{v}}_k{\mathbf{v}}_k^T)\Vert }^2 \\ \ge c_1{\sigma }_{{\min }^2}\left[-{\left({\mathbf{u}}^T{\mathbf{v}}_k\right)}^2+{\mathbf{b}}_{\mathbf{u},k,t_k}^T{\mathbf{b}}_{\mathbf{u},k,t_k}\right], \end{gathered}$$ (53)

where we have applied ${\mathbf{b}}_{\mathbf{u},k,t}^T{\mathbf{v}}_k={\mathbf{u}}^T\mathbf{v}$ again in the last step.

Let $\tilde{\mathbf{V}}=\left({\mathbf{v}}_{K+1},\cdots ,{\mathbf{v}}_n\right)$ be an n × (n – K) matrix such that $(\mathbf{V},\tilde{\mathbf{V}})$ is an orthogonal matrix of size n. Then the n-dimensional unit vector u can be represented as $\mathbf{u}={\sum }_{i=1}^n{a}_i{\mathbf{v}}_i$ for some scalars a_i’s. For each 1 ≤ k ≤ K, by the definition of $\mathcal{R}$ in (6) and Lemma 5 we can show that

$$\Vert \mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)+t_k^{-1}\mathbf{I}\Vert =O\left(\frac{{\alpha }_n^2}{{\left|t_k\right|}^3}\right)\text{and}\Vert \mathcal{R}\left(\mathbf{u},{\mathbf{V}}_{-k},t_k\right)+t_k^{-1}{\mathbf{u}}^T{\mathbf{V}}_{-k}\Vert =O\left(\frac{{\alpha }_n^2}{{\left|t_k\right|}^3}\right).$$ (54)

Therefore it holds that

$$\begin{gathered} \Vert \mathcal{R}\left(\mathbf{u},{\mathbf{V}}_{-k},t_k\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T+\sum _{1\le i\ne k\le K}{a}_i{\left(t_k{d}_i^{-1}-1\right)}^{-1}{\mathbf{v}}_i^T\Vert \\ =O\left({\alpha }_n^2/t_k^2\right). \end{gathered}$$ (55)

Then it follows from (55) and (8) that

$${\mathbf{b}}_{\mathbf{u},k,t_k}^T=\sum _{i=1}^n{a}_i{\mathbf{v}}_i^T-\mathcal{R}\left(\mathbf{u},{\mathbf{V}}_{-k},t_k\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T$$

and

$$\Vert {\mathbf{b}}_{\mathbf{u},k,t_k}-a_k{\mathbf{v}}_k-\sum _{1\le i\ne k\le K}{a}_i\left[1+{\left(t_k{d}_i^{-1}-1\right)}^{-1}\right]{\mathbf{v}}_i-\sum _{i=K+1}^n{a}_i{\mathbf{v}}_i\Vert =O\left({\alpha }_n^2/t_k^2\right).$$ (56)

We denote by ${\mathbf{c}}_k=a_k{\mathbf{v}}_k+{\sum }_{1\le i\ne k\le K}{a}_i\left[1+{\left(t_k{d}_i^{-1}-1\right)}^{-1}\right]{\mathbf{v}}_i+{\sum }_{i=K+1}^n{a}_i{\mathbf{v}}_i$. By (56), we can obtain

$$\begin{gathered} -{\left({\mathbf{u}}^T{\mathbf{v}}_k\right)}^2+{\mathbf{b}}_{\mathbf{u},k,t_k}^T{\mathbf{b}}_{\mathbf{u},k,t_k}=-a_k^2+{\Vert {\mathbf{c}}_k\Vert }^2+{\Vert {\mathbf{b}}_{\mathbf{u},k,t_k}-{\mathbf{c}}_k\Vert }^2+2{\left({\mathbf{b}}_{\mathbf{u},k,t_k}-{\mathbf{c}}_k\right)}^T{\mathbf{c}}_k \\ =\sum _{1\le i\ne k\le K}{a}_i^2{\left[1+{\left(t_k{d}_i^{-1}-1\right)}^{-1}\right]}^2+\sum _{i=K+1}^n{a}_i^2+O\left({\alpha }_n^2/t_k^2\right) \\ \text{+}\text{some}\text{small}\text{order}\text{term,} \end{gathered}$$ (57)

where the small order term takes a rather complicated form and thus we omit its expression for simplicity. Since by assumption $\left|{\mathbf{u}}^T{\mathbf{v}}_k\right|\in [0,1-ϵ],\mathbf{u}={\sum }_{i=1}^n{a}_i{\mathbf{v}}_i$ is a unit vector, and (v₁, · · ·, v_n) is an orthogonal matrix, it holds that

$$\sum _{1\le i\ne k\le n}^n{a}_i^2\ge 1-{(1-ϵ)}^2.$$ (58)

Moreover, Condition 3 and Lemma 3 together entail that $\left|t_k{d}_i^{-1}\right|$ is bounded away from 0 and 1. Thus there exists some positive constant c₂ < 1 such that

$${\left[1+{\left(t_k{d}_i^{-1}-1\right)}^{-1}\right]}^2\ge c_2$$ (59)

for each 1 ≤ i ≠ k ≤ K. Therefore, combining (53) and (57)–(59), and by the assumption ${\sigma }_{\min }\gg {\alpha }_n/t_k$, we can obtain the desired claim in (51), which completes the proof of Theorem 3.

A.4. Proof of Corollary 1

The conclusions of Corollary 1 follow directly from the results of Theorem 3.

A.5. Proof of Theorem 4

The more general asymptotic theory in Theorem 4 focuses on the first order asymptotic expansion for the bilinear form ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}$ with x and y two arbitrary n-dimensional unit vectors, while that in Theorem 5 further establishes the higher order (which is second order) asymptotic expansion for the same bilinear form. We begin with the analysis for the first order asymptotic expansion. The main ingredients of the proof are as follows. First, we represent ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}$ as an integral which is a functional of X = H + W. By doing so we can deal with the matrix H + W instead of the eigenvectors. Second, for the functional of H+W obtained in the previous step we extract the H part from H+W and further obtain a functional of W. Roughly speaking, we can get an explicit function of form f((W–tI)⁻¹) with |t| ≫ ‖W‖. Third, by the matrix series expansion ${(\mathbf{W}-t\mathbf{I})}^{-1}=-{\sum }_{l=0}^{\infty }{t}^{-(l+1)}{\mathbf{W}}^l$, the function f((W – tI)⁻¹) can be approximated by $f\left(-{\sum }_{l=0}^L{t}^{-(l+1)}{\mathbf{W}}^l\right)$ for some positive integer L. Fourth, we can then calculate the first (second or higher) order expansion of $f\left(-{\sum }_{l=0}^L{t}^{-(l+1)}{\mathbf{W}}^l\right)$ since we have an explicit expression of function f.

To facilitate our technical derivations, let us recall some basic matrix identities from the Sherman–Morrison–Woodbury formula. For any matrices A, B, C, and F of appropriate dimensions and any vectors a and b of appropriate dimensions, it holds that

$${(\mathbf{A}+\mathbf{B}\mathbf{F}\mathbf{C})}^{-1}={\mathbf{A}}^{-1}-{\mathbf{A}}^{-1}\mathbf{B}{\left({\mathbf{F}}^{-1}+\mathbf{C}{\mathbf{A}}^{-1}\mathbf{B}\right)}^{-1}\mathbf{C}{\mathbf{A}}^{-1}$$ (60)

and

$${\left(\mathbf{C}+\mathbf{a}{\mathbf{b}}^T\right)}^{-1}\mathbf{a}=\frac{{\mathbf{C}}^{-1}\mathbf{a}}{1+{\mathbf{b}}^T{\mathbf{C}}^{-1}\mathbf{a}}$$ (61)

when the corresponding matrices for matrix inversion are nonsingular.

To illustrate the main ideas of our proof, we first consider the simple case of K = 1 and x = y = v₁. The general case of K ≥ 1 and arbitrary unit vectors will be discussed later. Let Ω₁ be a contour centered at (a₁ + b₁)/2 with radius |b₁ – a₁|/2, where the quantities a_k and b_k with 1 ≤ k ≤ K are defined in Section 3.2. Then it is seen that d₁ is enclosed by Ω₁. In view of Condition 2, Lemma 6, and Weyl’s inequality, we have

$$\left|{\lambda }_1-d_1\right|\le \Vert \mathbf{W}\Vert <\min \left\{\left|d_1-a_1\right|,\left|d_1-b_1\right|\right\}$$

and

$$\left|{\lambda }_j-d_1\right|\ge \left|d_1\right|-\Vert \mathbf{W}\Vert >\max \left\{\left|d_1-a_1\right|,\left|d_1-b_1\right|\right\},j\ge 2$$

with significant probability. We can see that the contour Ω₁ does not enclose any other eigenvalues λ_j with j ≠ 1. Thus, by Cauchy’s residue theorem from complex analysis, we have with significant probability

$$-\frac{1}{2\pi i}{\oint }_{{\Omega }_1}\frac{1}{{\lambda }_1-z}dz=1\text{and}-\frac{1}{2\pi i}{\oint }_{{\Omega }_1}\frac{1}{{\lambda }_j-z}dz=0,j\ge 2,$$

where i associated with the complex integrals represents the imaginary unit (–1)^1/2 and the line integrals are taken over the contour Ω₁. Noticing that ${(\mathbf{X}-z\mathbf{I})}^{-1}={\sum }_{i=1}^n\sum _{i=1}^n{\left({\lambda }_j-z\right)}^{-1}{\widehat{\mathbf{v}}}_j{\widehat{\mathbf{v}}}_j^T$, we can then obtain an integral representation of the desired bilinear form that with significant probability

$$\begin{gathered} {\mathbf{v}}_1^T{\widehat{\mathbf{v}}}_1{\widehat{\mathbf{v}}}_1^T{\mathbf{v}}_1=-\frac{{\mathbf{v}}_1^T{\widehat{\mathbf{v}}}_1{\widehat{\mathbf{v}}}_1^T{\mathbf{v}}_1}{2\pi i}{\oint }_{{\Omega }_1}\frac{1}{{\lambda }_1-z}dz=-\frac{1}{2\pi i}{\oint }_{{\Omega }_1}{\mathbf{v}}_1^T\left(\sum _{j=1}^n\frac{{\widehat{\mathbf{v}}}_j{\widehat{\mathbf{v}}}_j^T}{{\lambda }_j-z}\right){\mathbf{v}}_{1}dz \\ =-\frac{1}{2\pi i}{\oint }_{{\Omega }_1}{\mathbf{v}}_1^T\tilde{\mathbf{G}}(z){\mathbf{v}}_{1}dz, \end{gathered}$$ (62)

where the matrix-valued function $\tilde{\mathbf{G}}(z)={(\mathbf{X}-z\mathbf{I})}^{-1}$ for z in the complex plane $ℂ$ is referred to as the Green function associated with the original random matrix X = H + W.

Note that by (1) and K = 1 for the simple case, we have $\mathbf{X}=\mathbf{H}+\mathbf{W}=d_1{\mathbf{v}}_1{\mathbf{v}}_1^T+\mathbf{W}$. Thus the line integral in (62) can be rewritten as

$${\mathbf{v}}_1^T{\widehat{\mathbf{v}}}_1{\widehat{\mathbf{v}}}_1^T{\mathbf{v}}_1=-\frac{1}{2\pi i}{\oint }_{{\Omega }_1}{\mathbf{v}}_1^T{\left(\mathbf{W}-z\mathbf{I}+d_1{\mathbf{v}}_1{\mathbf{v}}_1^T\right)}^{-1}{\mathbf{v}}_{1}dz.$$ (63)

With the aid of (60) and (61), the line integral in (63) can be further represented as

$${\mathbf{v}}_1^T{\widehat{\mathbf{v}}}_1{\widehat{\mathbf{v}}}_1^T{\mathbf{v}}_1=-\frac{1}{2\pi i}{\oint }_{{\Omega }_1}\frac{{\mathbf{v}}_1^T{(\mathbf{W}-z\mathbf{I})}^{-1}{\mathbf{v}}_1}{1+d_1{\mathbf{v}}_1^T{(\mathbf{W}-z\mathbf{I})}^{-1}{\mathbf{v}}_1}dz.$$ (64)

To analyze the integrand of the line integral on the right hand side of (64), we first consider the term (W – zI)⁻¹. Such a term admits the matrix series expansion

$${(\mathbf{W}-z\mathbf{I})}^{-1}=-\sum _{l=0}^{\infty }{z}^{-(l+1)}{\mathbf{W}}^l.$$ (65)

Let L be the smallest positive integer such that

$$\frac{{\alpha }_n^{L+1}{(\log n)}^{(L+1)/2}}{{\left|d_K\right|}^{L-2}}\to 0.$$ (66)

Such an integer L always exists since $\left|d_K\right|/\left(n^{ϵ}{\alpha }_n\right)\to \infty$ for small positive constant ϵ by Condition (2) and α_n ≤ n^1/2 by definition. Since we consider z on the contour Ω₁, it follows that |z| ≥ c|d₁| for some positive constant c. Thus, by (65), Condition 1, and Lemma 6 in Section B.7 of Supplementary Material, with the above choice of L in (66) we have with probability tending to one that

$$\Vert \sum _{l=L+1}^{\infty }{z}^{-(l+1)}{\mathbf{W}}^l\Vert \le \sum _{l=L+1}^{\infty }\frac{C^l{\alpha }_n^l{(\log n)}^{l/2}}{|z{|}^{l+1}}=\frac{O\left\{C^{L+1}{\alpha }_n^{L+1}{(\log n)}^{(L+1)/2}\right\}}{|z{|}^{L+2}}=\frac{O(1)}{|z{|}^4},$$ (67)

where C is some positive constant. In light of (65) and (67), we can obtain the asymptotic expansion

$$\begin{gathered} {\mathbf{v}}_1^T{\left(\mathbf{W}-z\mathbf{I}\right)}^{-1}{\mathbf{v}}_1=-\sum _{l=0}^L{z}^{-(l+1)}{\mathbf{v}}_1^T{\mathbf{W}}^l{\mathbf{v}}_1-\sum _{l=L+1}^{\infty }{z}^{-(l+1)}{\mathbf{v}}_1^T{\mathbf{W}}^l{\mathbf{v}}_1 \\ =-\sum _{l=0}^L{z}^{-(l+1)}{\mathbf{v}}_1^T{\mathbf{W}}^l{\mathbf{v}}_1+\frac{O_p(1)}{d_1^4} \end{gathered}$$ (68)

for z on the contour Ω₁.

Directly working with the line integral in (62) or (64) is challenging in deriving the CLT for the bilinear form ${\mathbf{v}}_1^T{\widehat{\mathbf{v}}}_1{\widehat{\mathbf{v}}}_1^T{\mathbf{v}}_1$. Next we introduce some simple facts about Cauchy’s residue theorem. Assume that a complex function f(z) is a holomorphic function inside Ω₁ except at one point t. Then it holds that

$$\frac{1}{2\pi i}{\oint }_{{\Omega }_1}f(z)dz=\mathrm{Res}(f,t),$$

where Res(f, t) represents the residue of function f at point t. In addition, assume that the Laurent series expansion of f around point t is given by

$$f(z)=\sum _{j=-\infty }^{\infty }{a}_j{(z-t)}^j$$

with a_j some constants. Then we have $\mathrm{Res}(f,t)={(2\pi i)}^{-1}{\oint }_{{\Omega }_1}f(z)dz=a_{-1}$. Furthermore, if lim_z→t(z – t)f(z) exists then the Laurent series expansion of f entails that

$$\underset{z\to t}{\lim }(z-t)f(z)=a_{-1}.$$ (69)

Now let us consider the line integral in (64). Observe that the only singular point of function ${\mathbf{v}}_1^T{(\mathbf{W}-z\mathbf{I})}^{-1}{\mathbf{v}}_1/\left[1+d_1{\mathbf{v}}_1^T{(\mathbf{W}-z\mathbf{I})}^{-1}{\mathbf{v}}_1\right]$ inside Ω₁ is the solution to equation

$$1+d_1{\mathbf{v}}_1^T{(\mathbf{W}-z\mathbf{I})}^{-1}{\mathbf{v}}_1=0,$$

which we denote as ${\widehat{t}}_1$. Let us use ${\left[{\left(\mathbf{W}-{\widehat{t}}_1\mathbf{I}\right)}^{-1}\right]}^{\prime }$ as a shorthand notation for $h^{\prime }\left({\widehat{t}}_1\right)$ with h(t) = (W – tI)⁻¹. Then by Cauchy’s residue theorem and in view of (64), we have

$$\begin{gathered} {\mathbf{v}}_1^T{\widehat{\mathbf{v}}}_1{\widehat{\mathbf{v}}}_1^T{\mathbf{v}}_1=-\frac{1}{2\pi i}{\oint }_{{\Omega }_1}\frac{{\mathbf{v}}_1^T{(\mathbf{W}-z\mathbf{I})}^{-1}{\mathbf{v}}_1}{1+d_1{\mathbf{v}}_1^T{(\mathbf{W}-z\mathbf{I})}^{-1}{\mathbf{v}}_1}dz=-\underset{z\to {\widehat{t}}_1}{\lim }\frac{\left(z-{\widehat{t}}_1\right){\mathbf{v}}_1^T{(\mathbf{W}-z\mathbf{I})}^{-1}{\mathbf{v}}_1}{1+d_1{\mathbf{v}}_1^T{(\mathbf{W}-z\mathbf{I})}^{-1}{\mathbf{v}}_1} \\ =-\frac{{\mathbf{v}}_1^T{\left(\mathbf{W}-{\widehat{t}}_1\mathbf{I}\right)}^{-1}{\mathbf{v}}_1}{d_1{\mathbf{v}}_1^T{\left[{\left(\mathbf{W}-{\widehat{t}}_1\mathbf{I}\right)}^{-1}\right]}^{\prime }{\mathbf{v}}_1}. \end{gathered}$$

Therefore, an application of the Taylor expansion to function ${\mathbf{v}}_1^T{\left(\mathbf{W}-{\widehat{t}}_1\mathbf{I}\right)}^{-1}{\mathbf{v}}_1/\left\{d_1{\mathbf{v}}_1^T{[(\mathbf{W}-{{\widehat{t}}_1\mathbf{I})}^{-1}]}^{\prime }{\mathbf{v}}_1\right\}$ yields

$$-\frac{{\mathbf{v}}_1^T{\left(\mathbf{W}-{\widehat{t}}_1\mathbf{I}\right)}^{-1}{\mathbf{v}}_1}{d_1{\mathbf{v}}_1^T{\left[{\left(\mathbf{W}-{\widehat{t}}_1\mathbf{I}\right)}^{-1}\right]}^{\prime }{\mathbf{v}}_1}=\frac{{\sum }_{l=1}^L{\widehat{t}}_1^{(l+1)}{\mathbf{v}}_1^T{\mathbf{W}}^l{\mathbf{v}}_1+d_1^{-4}{O}_p(1)}{d_1{\sum }_{l=0}^L(l+1){\widehat{t}}_1^{-(l+2)}{\mathbf{v}}_1^T{\mathbf{W}}^l{\mathbf{v}}_1+d_1^{-4}{O}_p(1)}.$$ (70)

Note that ${\widehat{t}}_1$ is a random variable that depends on random matrix X. In fact, from (99) we can see that the asymptotic expansion of ${\widehat{t}}_1$ is a polynomial of ${\mathbf{v}}_1^T{\mathbf{W}}^l{\mathbf{v}}_1$. Thus the asymptotic expansion of (70) is also a polynomial function of ${\mathbf{v}}_1^T{\mathbf{W}}^l{\mathbf{v}}_1$. Therefore, controlling the variance of ${\mathbf{v}}_1^T{\mathbf{W}}^l{\mathbf{v}}_1$ can facilitate us in identifying the leading term of the asymptotic expansion. So far we have laid out the major steps in deriving the asymptotic expansion for ${\mathbf{v}}_1^T{\widehat{\mathbf{v}}}_1{\widehat{\mathbf{v}}}_1^T{\mathbf{v}}_1$. This can shed light on the detailed proof for the general case of ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}$ with K ≥ 1.

We now move on to the general case of K ≥ 1 and arbitrary n-dimensional unit vectors x and y. The technical arguments for the general case are similar to those for the simple case of K = 1 and x = y = v₁ presented above, but with more delicate technical derivations. Similarly as in (62), it follows from Cauchy’s residue theorem, the definitions of the eigenvalue and eigenvector, and (1) that the bilinear form ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}$ for each 1 ≤ k ≤ K admits a natural integral representation; that is, with significant probability,

$$\begin{gathered} {\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}=-\frac{1}{2\pi i}{\oint }_{{\Omega }_k}{\mathbf{x}}^T\tilde{\mathbf{G}}(z)\mathbf{y}dz=-\frac{1}{2\pi i}{\oint }_{{\Omega }_k}{\mathbf{x}}^T{\left(\mathbf{W}-z\mathbf{I}+\sum _{j=1}^K{d}_j{\mathbf{v}}_j{\mathbf{v}}_j^T\right)}^{-1}\mathbf{y}dz \\ =\frac{1}{2\pi i}{\oint }_{{\Omega }_k}\frac{d_k{\mathbf{x}}^T{\left(\mathbf{W}-z\mathbf{I}+\sum _{1\le j\ne k\le K}{d}_j{\mathbf{v}}_j{\mathbf{v}}_j^T\right)}^{-1}{\mathbf{v}}_k{\mathbf{v}}_k^T{\left(\mathbf{W}-z\mathbf{I}+\sum _{1\le j\ne k\le K}{d}_i{\mathbf{v}}_i{\mathbf{v}}_i^T\right)}^{-1}\mathbf{y}}{1+d_k{\mathbf{v}}_k^T{\left(\mathbf{W}-z\mathbf{I}+\sum _{1\le j\ne k\le K}{d}_j{\mathbf{v}}_j{\mathbf{v}}_j^T\right)}^{-1}{\mathbf{v}}_k}dz, \end{gathered}$$ (71)

where the Green function $\tilde{\mathbf{G}}(z)$ associated with the original random matrix X is defined in (62) and the line integral is taken over a contour Ω_k that is centered at (a_k + b_k)/2 with radius |b_k – a_k|/2. Then the contour Ω_k encloses the population eigenvalue d_k of the latent mean matrix H. Note that in the representation above, we have used the results, which can be derived from Condition 2, Lemma 6, and Weyl’s inequality, that for each j = 1, · · ·, K,

$$\left|{\lambda }_k-d_k\right|\le \Vert \mathbf{W}\Vert <\min \left\{\left|d_k-a_k\right|,\left|d_k-b_k\right|\right\},$$

$$\left|{\lambda }_j-d_k\right|\ge \left|d_j-d_k\right|-\left|{\lambda }_j-d_j\right|\ge \left|d_j-d_k\right|-\Vert \mathbf{W}\Vert >\max \left\{\left|d_k-a_k\right|,\left|d_k-b_k\right|\right\}$$

for j ≠ k with significant probability; that is, the contour Ω_k encloses λ_k but not any other eigenvalues with high probability.

An application of (60) leads to

$${\left(\mathbf{W}-z\mathbf{I}+\sum _{1\le j\ne k\le K}{d}_j{\mathbf{v}}_j{\mathbf{v}}_j^T\right)}^{-1}=\mathbf{G}(z)-\mathbf{G}(z){\mathbf{V}}_{-k}{\left[{\mathbf{D}}_{-k}^{-1}+{\mathbf{V}}_{-k}^T\mathbf{G}(z){\mathbf{V}}_{-k}\right]}^{-1}{\mathbf{V}}_{-k}^T\mathbf{G}(z),$$ (72)

where the Green function G(z) associated with only the noise part W is defined in (37). To simplify the expression, let

$${\mathbf{F}}_k(z)=\mathbf{G}(z){\mathbf{V}}_{-k}{\left[{\mathbf{D}}_{-k}^{-1}+{\mathbf{V}}_{-k}^T\mathbf{G}(z){\mathbf{V}}_{-k}\right]}^{-1}{\mathbf{V}}_{-k}^T\mathbf{G}(z)$$ (73)

Then in view of (72), the last line integral in (71) can be further represented as

$${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}=\frac{1}{2\pi i}{\oint }_{{\Omega }_k}\frac{d_k{\mathbf{x}}^T\left[\mathbf{G}(z)-{\mathbf{F}}_k(z)\right]{\mathbf{v}}_k{\mathbf{v}}_k^T\left[\mathbf{G}(z)-{\mathbf{F}}_k(z)\right]\mathbf{y}}{1+d_k{\mathbf{v}}_k^T\left[\mathbf{G}(z)-{\mathbf{F}}_k(z)\right]{\mathbf{v}}_k}dz.$$ (74)

It is challenging to analyze the terms in (74) since the expression of F_k(z) is complicated and we need to study the asymptotic expansion of F_k(z) carefully. In the proof below, we will see that Lemma 4 in Section 6 is a key ingredient of the technical arguments; see Section B.5 of Supplementary Material for the proof of this lemma.

We will conduct detailed calculations for the asymptotic expansion of F_k(z). Let us choose L as the same positive integer as in (66). Then we have ${\sum }_{l=L+1}^{\infty }{z}^{-(l+1)}{\mathbf{x}}^T{\mathbf{W}}^l\mathbf{y}=O_p\left(|z{|}^{-4}\right)$ for z on the contour Ω_k. It follows from Lemma 4 and Condition 2 that

$$\begin{gathered} \sum _{l=2}^L{z}^{-(l+1)}{\mathbf{x}}^T\left({\mathbf{W}}^l-\mathbb{E}{\mathbf{W}}^l\right)\mathbf{y}=O_p\left\{{\alpha }_n|z{|}^{-3}+{\alpha }_n^2|z{|}^{-4}+\cdots +{\alpha }_n^{L-1}|z{|}^{-(L+1)}\right\} \\ =O_p\left({\alpha }_n|z{|}^{-3}\right). \end{gathered}$$

Therefore, similar to (68) we can show that

$$\begin{gathered} {\mathbf{x}}^T\mathbf{G}(z)\mathbf{y}=-z^{-1}{\mathbf{x}}^T\mathbf{y}-z^{-2}{\mathbf{x}}^T\mathbf{W}\mathbf{y}-\sum _{l=2}^L{z}^{-(l+1)}{\mathbf{x}}^T\mathbb{E}{\mathbf{W}}^l\mathbf{y}-\sum _{l=L+1}^{\infty }{z}^{-(l+1)}{\mathbf{x}}^T{\mathbf{W}}^l\mathbf{y} \\ -\sum _{l=2}^L{z}^{-(l+1)}{\mathbf{x}}^T\left({\mathbf{W}}^l-\mathbb{E}{\mathbf{W}}^l\right)\mathbf{y} \\ =-z^{-1}{\mathbf{x}}^T\mathbf{y}-z^{-2}{\mathbf{x}}^T\mathbf{W}\mathbf{y}-\sum _{l=2}^L{z}^{-(l+1)}{\mathbf{x}}^T\mathbb{E}{\mathbf{W}}^l\mathbf{y}+O_p\left(|z{|}^{-4}+{\alpha }_n|z{|}^{-3}\right). \end{gathered}$$ (75)

Moreover, since for z ∈ Ω_k we have |z|^–4 ≤ α_n|z|⁻³ by Condition 1, we can further obtain

$${\mathbf{x}}^T\mathbf{G}(z)\mathbf{y}=-z^{-1}{\mathbf{x}}^T\mathbf{y}-z^{-2}{\mathbf{x}}^T\mathbf{W}\mathbf{y}-\sum _{l=2}^L{z}^{-(l+1)}{\mathbf{x}}^T\mathbb{E}{\mathbf{W}}^l\mathbf{y}+O_p\left({\alpha }_n|z{|}^{-3}\right).$$ (76)

In fact, the probabilistic event associated with the small order term O_p(α_n|z|^–3) in (76) holds uniformly over z since the term O_p(α_n|z|⁻³) is simply |z|⁻³O_p(α_n).

To simplify the technical presentation, hereafter we use the generic notation u to denote either x or y unless specified otherwise, which means that the corresponding derivations and results hold when u is replaced by x and y. Since x and y can be chosen as any unit vectors, we can obtain from (76) the following asymptotic expansions by different choices of x and y

$${\mathbf{u}}^T\mathbf{G}(z){\mathbf{v}}_k=-z^{-1}{\mathbf{u}}^T{\mathbf{v}}_k-z^{-2}{\mathbf{u}}^T\mathbf{W}{\mathbf{v}}_k-\sum _{l=2}^L{z}^{-(l+1)}{\mathbf{u}}^T\mathbb{E}{\mathbf{W}}^l{\mathbf{v}}_k+O_p\left({\alpha }_n|z{|}^{-3}\right),$$ (77)

$${\mathbf{v}}_k^T\mathbf{G}(z){\mathbf{v}}_k=-z^{-1}-z^{-2}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k-\sum _{l=2}^L{z}^{-(l+1)}{\mathbf{v}}_k^T\mathbb{E}{\mathbf{W}}^l{\mathbf{v}}_k+O_p\left({\alpha }_n|z{|}^{-3}\right),$$ (78)

$${\mathbf{v}}_k^T\mathbf{G}(z){\mathbf{V}}_{-k}=-z^{-2}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{V}}_{-k}-\sum _{l=2}^L{z}^{-(l+1)}{\mathbf{v}}_k^T\mathbb{E}{\mathbf{W}}^l{\mathbf{V}}_{-k}+O_p\left({\alpha }_n|z{|}^{-3}\right),$$ (79)

$$\begin{gathered} {\mathbf{u}}^T\mathbf{G}(z){\mathbf{V}}_{-k}=-z^{-1}{\mathbf{u}}^T{\mathbf{V}}_{-k}-z^{-2}{\mathbf{u}}^T\mathbf{W}{\mathbf{V}}_{-k}-\sum _{l=2}^L{z}^{-(l+1)}{\mathbf{u}}^T\mathbb{E}{\mathbf{W}}^l{\mathbf{V}}_{-k} \\ +O_p\left({\alpha }_n|z{|}^{-3}\right), \end{gathered}$$ (80)

$${\mathbf{V}}_{-k}^T\mathbf{G}(z){\mathbf{V}}_{-k}=-z^{-1}\mathbf{I}-z^{-2}{\mathbf{V}}_{-k}^T\mathbf{W}{\mathbf{V}}_{-k}-\sum _{l=2}^L{z}^{-(l+1)}{\mathbf{V}}_{-k}^T\mathbb{E}{\mathbf{W}}^l{\mathbf{V}}_{-k}+O_p\left({\alpha }_n|z{|}^{-3}\right).$$ (81)

Thus it follows from (76)–(81) that

$$\begin{gathered} {\mathbf{u}}^T{\mathbf{F}}_k(z){\mathbf{v}}_k=\mathcal{R}\left(\mathbf{u},{\mathbf{V}}_{-k},z\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{v}}_k,z\right) \\ -z^{-2}\mathcal{R}\left(\mathbf{u},{\mathbf{V}}_{-k},z\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\mathbf{W}{\mathbf{v}}_k \\ -z^{-2}{\mathbf{u}}^T\mathbf{W}{\mathbf{V}}_{-k}{\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{v}}_k,z\right) \\ +z^{-2}\mathcal{R}\left(\mathbf{u},{\mathbf{V}}_{-k},z\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\mathbf{W}{\mathbf{V}}_{-k} \\ =\mathcal{R}\left(\mathbf{u},{\mathbf{V}}_{-k},z\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\mathbf{W}{\mathbf{V}}_{-k} \\ -z^{-2}\mathcal{R}\left(\mathbf{u},{\mathbf{V}}_{-k},z\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n|z{|}^{-3}\right) \end{gathered}$$ (82)

and

$$\begin{gathered} {\mathbf{v}}_k^T{\mathbf{F}}_k(z){\mathbf{v}}_k={\mathbf{v}}_k^T\mathbf{G}(z){\mathbf{V}}_{-k}{\left[{\mathbf{D}}_{-k}^{-1}+{\mathbf{V}}_{-k}^T\mathbf{G}(z){\mathbf{V}}_{-k}\right]}^{-1}{\mathbf{V}}_{-k}^T\mathbf{G}(z){\mathbf{v}}_k \\ =\mathcal{R}\left({\mathbf{v}}_k,{\mathbf{V}}_{-k},z\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{v}}_k,z\right) \\ -z^{-2}\mathcal{R}\left({\mathbf{v}}_k,{\mathbf{V}}_{-k},z\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n|z{|}^{-3}\right) \\ =\mathcal{R}\left({\mathbf{v}}_k,{\mathbf{V}}_{-k},z\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{v}}_k,z\right)+O_p\left({\alpha }_n|z{|}^{-3}\right), \end{gathered}$$ (83)

where F_k(z) is defined in (73) and $\mathcal{R}$ is defined in (6).

With all the technical preparations above, we are now ready to analyze the terms in representation (74). Specifically let us consider the ratio $\{d_k{\mathbf{x}}^T\left[\mathbf{G}(z)-{\mathbf{F}}_k(z)\right]{\mathbf{v}}_k{\mathbf{v}}_k^T[\mathbf{G}(z)-{\mathbf{F}}_k(z)]\mathbf{y}\}/\left\{1+d_k{\mathbf{v}}_k^T\left[\mathbf{G}(z)-{\mathbf{F}}_k(z)\right]{\mathbf{v}}_k\right\}$ that appears as the integrand on the left hand side of (74). Similar to (75), taking the derivative of G(z) we have

$$\begin{gathered} {\mathbf{x}}^T{\mathbf{G}}^{\prime }(z)\mathbf{y}={\mathbf{x}}^T{\left(\mathbf{W}-z\mathbf{I}\right)}^{-2}\mathbf{y}=\sum _{l=0}^{\infty }(l+1)z^{-(l+2)}{\mathbf{x}}^T{\mathbf{W}}^l\mathbf{y} \\ ={\mathcal{R}}^{\prime }(\mathbf{x},\mathbf{y},z)+2z^{-3}{\mathbf{x}}^T\mathbf{W}y+z^{-4}{O}_p\left({\alpha }_n\right). \end{gathered}$$ (84)

It follows from Lemmas 4–5 that

$$\begin{gathered} {\mathcal{R}}^{\prime }\left({\mathbf{v}}_k,{\mathbf{V}}_{-k},z\right)=O\left({\alpha }_n^2/z^4\right),{\mathcal{R}}^{\prime }\left({\mathbf{v}}_k,{\mathbf{v}}_k,z\right)-\frac{1}{z^2}=O\left({\alpha }_n^2/z^4\right), \\ \Vert {\mathcal{R}}^{\prime }\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)-z^{-2}\mathbf{I}\Vert =O\left({\alpha }_n^2/z^4\right) \end{gathered}$$ (85)

By (79) and Lemmas 4–5, we can conclude that

$${\mathbf{v}}_k^T\mathbf{G}(z){\mathbf{V}}_{-k}=z^{-2}{O}_p(1)+|z{|}^{-3}{O}_p\left({\alpha }_n^2\right).$$ (86)

Moreover, by (80) and (A.16) we have

$$\begin{gathered} \Vert {\left\{{\left[{\mathbf{D}}_{-k}^{-1}+{\mathbf{V}}_{-k}^T\mathbf{G}(z){\mathbf{V}}_{-k}\right]}^{-1}-{\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}\right\}}^{\prime }\Vert \\ =\Vert {\left[{\mathbf{D}}_{-k}^{-1}+{\mathbf{V}}_{-k}^T\mathbf{G}(z){\mathbf{V}}_{-k}\right]}^{-1}{\mathbf{V}}_{-k}^T{\mathbf{G}}^{\prime }(z){\mathbf{V}}_{-k}{\left[{\mathbf{D}}_{-k}^{-1}+{\mathbf{V}}_{-k}^T\mathbf{G}(z){\mathbf{V}}_{-k}\right]}^{-1} \\ -{\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}{\mathcal{R}}^{\prime }\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}\Vert \\ =O\left\{\Vert {\mathbf{V}}_{-k}^T{\mathbf{G}}^{\prime }(z){\mathbf{V}}_{-k}-{\mathcal{R}}^{\prime }\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\Vert {\Vert {\left[{\mathbf{D}}_{-k}^{-1}+{\mathbf{V}}_{-k}^T\mathbf{G}(z){\mathbf{V}}_{-k}\right]}^{-1}\Vert }^2\right\} \\ +O\{\Vert {\left[{\mathbf{D}}_{-k}^{-1}+{\mathbf{V}}_{-k}^T\mathbf{G}(z){\mathbf{V}}_{-k}\right]}^{-1}-{\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}\Vert \\ \cdot \Vert {\left[{\mathbf{D}}_{-k}^{-1}+{\mathbf{V}}_{-k}^T\mathbf{G}(z){\mathbf{V}}_{-k}\right]}^{-1}{\mathcal{R}}^{\prime }\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\Vert \} \\ =|z{|}^{-1}{O}_p(1)+z^{-2}{O}_p\left({\alpha }_n\right) \end{gathered}$$ (87)

and

$$\begin{gathered} \Vert {\left\{{\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}\right\}}^{\prime }\Vert \\ =\Vert {\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}{\mathcal{R}}^{\prime }\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}\Vert \\ \text{=}O\text{(1)}. \end{gathered}$$ (88)

Note that in light of (84)–(87), we can obtain

$$\begin{gathered} {\mathbf{v}}_k^T{\mathbf{F}}_k^{\prime }(z){\mathbf{v}}_k=2{\mathbf{v}}_k^T{\mathbf{G}}^{\prime }(z){\mathbf{V}}_{-k}{\left[{\mathbf{D}}_{-k}^{-1}+{\mathbf{V}}_{-k}^T\mathbf{G}(z){\mathbf{V}}_{-k}\right]}^{-1}{\mathbf{V}}_{-k}^T\mathbf{G}(z){\mathbf{v}}_k \\ +{\mathbf{v}}_k^T\mathbf{G}(z){\mathbf{V}}_{-k}{\left\{{\left[{\mathbf{D}}_{-k}^{-1}+{\mathbf{V}}_{-k}^T\mathbf{G}(z){\mathbf{V}}_{-k}\right]}^{-1}\right\}}^{\prime }{\mathbf{V}}_{-k}^T\mathbf{G}(z){\mathbf{v}}_k \\ =2{\mathcal{R}}^{\prime }\left({\mathbf{v}}_k,{\mathbf{V}}_{-k},z\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{v}}_k,z\right) \\ +\mathcal{R}\left({\mathbf{v}}_k,{\mathbf{V}}_{-k},z\right){\left\{{\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}\right\}}^{\prime }\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{v}}_k,z\right) \\ +z^{-4}{O}_p(1)+z^{-6}{O}_p\left({\alpha }_n^3\right). \end{gathered}$$ (89)

Combining the above result with (84) leads to

$$d_k{\mathbf{v}}_k^T\left[{\mathbf{G}}^{\prime }(z)-{\mathbf{F}}_k^{\prime }(z)\right]{\mathbf{v}}_k=\frac{d_k}{z^2{\tilde{\mathcal{P}}}_{k,z}}+2z^{-3}{d}_k{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k+z^{-4}{O}_p\left(\left|d_k\right|{\alpha }_n\right)$$ (90)

for z ∈ [a_k, b_k]. Further, recalling the definition in (7) and by (88), it holds that

$$\begin{gathered} \frac{1}{z^2{\tilde{\mathcal{P}}}_{k,z}}={\left(\frac{A_{{\mathbf{v}}_k,k,z}}{z}\right)}^{\prime }={\mathcal{R}}^{\prime }\left({\mathbf{v}}_k,{\mathbf{v}}_k,z\right)-2{\mathcal{R}}^{\prime }\left({\mathbf{v}}_k,{\mathbf{V}}_{-k},z\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1} \\ \times \mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{v}}_k,z\right)-\mathcal{R}\left({\mathbf{v}}_k,{\mathbf{V}}_{-k},z\right){\left\{{\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},z\right)\right]}^{-1}\right\}}^{\prime }\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{v}}_k,z\right) \\ =z^{-2}+O\left({\alpha }_n^2/z^4\right). \end{gathered}$$ (91)

Plugging this into (90) and by Lemmas 4–5, we have for all z ∈ [a_k, b_k],

$$\begin{gathered} d_k{\mathbf{v}}_k^T\left[{\mathbf{G}}^{\prime }(z)-{\mathbf{F}}_k^{\prime }(z)\right]{\mathbf{v}}_k=d_k{z}^{-2}+2z^{-3}{d}_k{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k+z^{-4}{O}_p\left(\left|d_k\right|{\alpha }_n^2\right) \\ =d_k{z}^{-2}\left[1+O_p\left(|z{|}^{-1}+|z{|}^{-2}{\alpha }_n^2\right)\right]=d_k{z}^{-2}\left[1+o_p(1)\right]. \end{gathered}$$ (92)

Thus $1+d_k{\mathbf{v}}_k^T\left[\mathbf{G}(z)-{\mathbf{F}}_k(z)\right]{\mathbf{v}}_k$ is a monotone function with probability tending to one.

Further, in light of expressions (78) and (83) we can obtain the asymptotic expansion

$$1+d_k{\mathbf{v}}_k^T\left[\mathbf{G}(z)-{\mathbf{F}}_k(z)\right]{\mathbf{v}}_k=f_k(z)-d_k{z}^{-2}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k+z^{-2}{O}_p\left({\alpha }_n\right)$$ (93)

for all z ∈ [a_k,b_k], where f_k(z) is defined in (10). Note that f_k(a_k) = O(1), f_k(b_k) = O(1), and f_k(a_k)f_k(b_k) < 0 as shown in the proof of Lemma 3 in Section B.4 of Supplementary Material. These results together with (92), which gives the order for the derivative of $1+d_k{\mathbf{v}}_k^T\left[\mathbf{G}(z)-{\mathbf{F}}_k(z)\right]{\mathbf{v}}_k$, entail that there exists a unique solution ${\widehat{t}}_k$ to the equation

$$1+d_k{\mathbf{v}}_k^T\left[\mathbf{G}(z)-{\mathbf{F}}_k(z)\right]{\mathbf{v}}_k=0$$ (94)

for z in the interval [a_k, b_k]. Using Lemma 4, we can further show that (93) becomes

$$1+d_k{\mathbf{v}}_k^T\left[\mathbf{G}(z)-{\mathbf{F}}_k(z)\right]{\mathbf{v}}_k-f_k(z)=-\frac{d_k}{z^2}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k+O_p\left(|z{|}^{-2}{\alpha }_n\right)=O_p\left(|z{|}^{-1}\right)$$ (95)

for z ∈ [a_k, b_k]. Note that f_k(z) is a monotone function over z ∈ [a_k, b_k] as shown in the proof of Lemma 3 and (A.17). Thus it follows from (94) and (95) that

$${\widehat{t}}_k-t_k=O_p(1).$$ (96)

In fact, we can obtain a more precise order of ${\widehat{t}}_k-t_k$ than the initial one in (96). In view of (93) and the definition of t_k, we have

$$1+d_k{\mathbf{v}}_k^T\left[\mathbf{G}\left(t_k\right)-{\mathbf{F}}_k\left(t_k\right)\right]{\mathbf{v}}_k=-d_k{t}_k^{-2}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n{t}_k^{-2}\right).$$ (97)

By (92) and (97), an application of the mean value theorem yields

$$\begin{gathered} 0=1+d_k{\mathbf{v}}_k^T\left[\mathbf{G}\left({\widehat{t}}_k\right)-{\mathbf{F}}_k\left({\widehat{t}}_k\right)\right]{\mathbf{v}}_k=1+d_k{\mathbf{v}}_k^T\left[\mathbf{G}\left(t_k\right)-{\mathbf{F}}_k\left(t_k\right)\right]{\mathbf{v}}_k \\ +d_k{\tilde{t}}_k^{-2}\left[1+O_p\left({\left|d_k\right|}^{-1}+{\left|d_k\right|}^{-2}{\alpha }_n^2\right)\right]\left({\widehat{t}}_k-t_k\right), \end{gathered}$$ (98)

where ${\tilde{t}}_k$ is some number between t_k and ${\widehat{t}}_k$. The asymptotic expansions in (98) and (97) entail further that

$${\widehat{t}}_k-t_k=\frac{{\mathfrak{t}}_k^2}{t_k^2}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n{t}_k^{-1}\right)={\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n{t}_k^{-1}\right).$$ (99)

Now by the similar arguments as for obtaining (69), the integral (74) can be evaluated as

$$\begin{gathered} {\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}=\frac{1}{2\pi i}{\oint }_{{\Omega }_k}\frac{d_k{\mathbf{x}}^T\left[\mathbf{G}(z)-{\mathbf{F}}_k(z)\right]{\mathbf{v}}_k{\mathbf{v}}_k^T\left[\mathbf{G}(z)-{\mathbf{F}}_k(z)\right]\mathbf{y}}{1+d_k{\mathbf{v}}_k^T\left[\mathbf{G}(z)-{\mathbf{F}}_k(z)\right]{\mathbf{v}}_k}dz \\ =\frac{{\widehat{t}}_k^2{\mathbf{x}}^T\left[\mathbf{G}\left({\widehat{t}}_k\right)-{\mathbf{F}}_k\left({\widehat{t}}_k\right)\right]{\mathbf{v}}_k{\mathbf{v}}_k^T\left[\mathbf{G}\left({\widehat{t}}_k\right)-{\mathbf{F}}_k\left({\widehat{t}}_k\right)\right]\mathbf{y}}{{\widehat{t}}_k^2{\mathbf{v}}_k^T\left[{\mathbf{G}}^{\prime }\left({\widehat{t}}_k\right)-{\mathbf{F}}_k^{\prime }\left({\widehat{t}}_k\right)\right]{\mathbf{v}}_k}. \end{gathered}$$ (100)

By (90) we have

$$\frac{1}{{\widehat{t}}_k^2{\mathbf{v}}_k^T\left[{\mathbf{G}}^{\prime }\left({\widehat{t}}_k\right)-{\mathbf{F}}_k^{\prime }\left({\widehat{t}}_k\right)\right]{\mathbf{v}}_k}={\tilde{\mathcal{P}}}_{k,{\widehat{t}}_k}-2{\widehat{t}}_k^{-1}{\tilde{\mathcal{P}}}_{k,{\widehat{t}}_k}^2{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k+{\widehat{t}}_k^{-2}{O}_p\left({\alpha }_n\right)$$ (101)

and (100) can be written as

$$\begin{gathered} {\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}=\frac{{\widehat{t}}_k^2{\mathbf{x}}^T\left[\mathbf{G}\left({\widehat{t}}_k\right)-{\mathbf{F}}_k\left({\widehat{t}}_k\right)\right]{\mathbf{v}}_k{\mathbf{v}}_k^T\left[\mathbf{G}\left({\widehat{t}}_k\right)-{\mathbf{F}}_k\left({\widehat{t}}_k\right)\right]\mathbf{y}}{{\widehat{t}}_k^2{\mathbf{v}}_k^T\left[{\mathbf{G}}^{\prime }\left({\widehat{t}}_k\right)-{\mathbf{F}}_k^{\prime }\left({\widehat{t}}_k\right)\right]{\mathbf{v}}_k} \\ =\left[{\tilde{\mathcal{P}}}_{k,{\widehat{t}}_k}-2{\widehat{t}}_k^{-1}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k+{\widehat{t}}_k^{-2}{O}_p\left({\alpha }_n\right)\right]{\widehat{t}}_k^2{\mathbf{x}}^T\left[\mathbf{G}\left({\widehat{t}}_k\right)-{\mathbf{F}}_k\left({\widehat{t}}_k\right)\right]{\mathbf{v}}_k{\mathbf{v}}_k^T \\ \times \left[\mathbf{G}\left({\widehat{t}}_k\right)-{\mathbf{F}}_k\left({\widehat{t}}_k\right)\right]\mathbf{y}. \end{gathered}$$ (102)

Recall the definitions in (6) and (7). Then it follows from (77), (82), and (99) that

$$\begin{gathered} {\widehat{t}}_k{\mathbf{u}}^T\left[\mathbf{G}\left({\widehat{t}}_k\right)-{\mathbf{F}}_k\left({\widehat{t}}_k\right)\right]{\mathbf{v}}_k=\mathcal{P}\left(\mathbf{u},{\mathbf{v}}_k,{\widehat{t}}_k\right)-\mathcal{P}\left(\mathbf{u},{\mathbf{V}}_{-k},{\widehat{t}}_k\right){\left[{\widehat{t}}_k{\mathbf{D}}_{-k}^{-1}+\mathcal{P}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},{\widehat{t}}_k\right)\right]}^{-1} \\ \times \mathcal{P}\left({\mathbf{V}}_{-k},{\mathbf{v}}_k,{\widehat{t}}_k\right)-{\widehat{t}}_k^{-1}{\mathbf{u}}^T\mathbf{W}{\mathbf{v}}_k+{\widehat{t}}_k^{-1}\mathcal{R}\left(\mathbf{u},{\mathbf{V}}_{-k},{\widehat{t}}_k\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},{\widehat{t}}_k\right)\right]}^{-1} \\ \times {\mathbf{V}}_{-k}^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n{\widehat{t}}_k^{-2}\right) \\ =\mathcal{P}\left(\mathbf{u},{\mathbf{v}}_k,t_k\right)-\mathcal{P}\left(\mathbf{u},{\mathbf{V}}_{-k},t_k\right){\left[t_k{\mathbf{D}}_{-k}^{-1}+\mathcal{P}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1} \\ \times \mathcal{P}\left({\mathbf{V}}_{-k},{\mathbf{v}}_k,t_k\right)-t_k^{-1}{\mathbf{u}}^T\mathbf{W}{\mathbf{v}}_k+t_k^{-1}\mathcal{R}\left(\mathbf{u},{\mathbf{V}}_{-k},t_k\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1} \\ \times {\mathbf{V}}_{-k}^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n{t}_k^{-2}\right) \\ =A_{\mathbf{u},k,t_k}-t_k^{-1}{\mathbf{b}}_{\mathbf{u},k,t_k}^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n{t}_k^{-2}\right), \end{gathered}$$ (103)

where u stands for both x and y as mentioned before. Furthermore, by Lemma 5 and (99) we can conclude that

$${\tilde{\mathcal{P}}}_{k,\widehat{t_k}}={\tilde{\mathcal{P}}}_{k,t_k}+O_p\left({\alpha }_n^2{t}_k^{-3}\right).$$ (104)

Combining the representation (102) and asymptotic expansions (103)–(104), by Lemma 4 we can deduce that (100) can be further written as

$$\begin{gathered} {\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}=\frac{{\widehat{t}}_k^2{\mathbf{x}}^T\left[\mathbf{G}\left({\widehat{t}}_k\right)-{\mathbf{F}}_k\left({\widehat{t}}_k\right)\right]{\mathbf{v}}_k{\mathbf{v}}_k^T\left[\mathbf{G}\left({\widehat{t}}_k\right)-{\mathbf{F}}_k\left({\widehat{t}}_k\right)\right]\mathbf{y}}{{\widehat{t}}_k^2{\mathbf{v}}_k^T\left[{\mathbf{G}}^{\prime }\left({\widehat{t}}_k\right)-{\mathbf{F}}_k^{\prime }\left({\widehat{t}}_k\right)\right]{\mathbf{v}}_k} \\ =\left[{\tilde{\mathcal{P}}}_{k,t_k}-2t_k^{-1}{\tilde{\mathcal{P}}}_{k,t_k}^2{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n{t}_k^{-2}\right)\right]\left[A_{\mathbf{x},k,t_k}-t_k^{-1}{\mathbf{b}}_{\mathbf{x},k,t_k}^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n{t}_k^{-2}\right)\right] \\ \times \left[A_{\mathbf{y},k,t_k}-t_k^{-1}{\mathbf{b}}_{\mathbf{y},k,t_k}^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n{t}_k^{-2}\right)\right] \\ =\left[{\tilde{\mathcal{P}}}_{k,t_k}-2t_k^{-1}{\tilde{\mathcal{P}}}_{k,t_k}^2{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n{t}_k^{-2}\right)\right]\times [A_{\mathbf{x},k,t_k}{A}_{\mathbf{y},k,t_k} \\ -t_k^{-1}\left(A_{\mathbf{x},k,t_k}{\mathbf{b}}_{\mathbf{x},k,t_k}^T+A_{\mathbf{y},k,t_k}{\mathbf{b}}_{\mathbf{y},k,t_k}^T\right)\mathbf{W}{\mathbf{v}}_k \\ +t_k^{-2}{\mathbf{b}}_{\mathbf{x},k,t_k}^T\mathbf{W}{\mathbf{v}}_k{\mathbf{b}}_{\mathbf{y},k,t_k}^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n{c}_k{t}_k^{-2}\right)], \end{gathered}$$ (105)

where $c_k=\left|A_{\mathbf{x},k,t_k}\right|+\left|A_{\mathbf{y},k,t_k}\right|+{\left|t_k\right|}^{-1}$.

We can expand (105), or equivalently (100), further as

$$\begin{gathered} {\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}=\left[{\tilde{\mathcal{P}}}_{k,t_k}-2t_k^{-1}{\tilde{\mathcal{P}}}_{k,t_k}^2{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n{t}_k^{-2}\right)\right]\times [A_{\mathbf{x},k,t_k}{A}_{\mathbf{y},k,t_k} \\ -t_k^{-1}\left(A_{\mathbf{x},k,t_k}{\mathbf{b}}_{\mathbf{x},k,t_k}^T+A_{\mathbf{y},k,t_k}{\mathbf{b}}_{\mathbf{y},k,t_k}^T\right)\mathbf{W}{\mathbf{v}}_k+t_k^{-2}{\mathbf{b}}_{\mathbf{x},k,t_k}^T\mathbf{W}{\mathbf{v}}_k{\mathbf{b}}_{\mathbf{y},k,t_k}^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n{c}_k{t}_k^{-2}\right)] \\ ={\tilde{\mathcal{P}}}_{k,t_k}{A}_{\mathbf{x},k,t_k}{A}_{\mathbf{y},k,t_k}-t_k^{-1}{A}_{\mathbf{x},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}({\mathbf{b}}_{\mathbf{y},k,t_k}^T+A_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k \\ -t_k^{-1}{A}_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}({\mathbf{b}}_{\mathbf{x},k,t_k}^T+A_{\mathbf{x},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k \\ +t_k^{-2}{\tilde{\mathcal{P}}}_{k,t_k}\left[2{\tilde{\mathcal{P}}}_{k,t_k}\left(A_{\mathbf{x},k,t_k}{\mathbf{b}}_{\mathbf{x},k,t_k}^T+A_{\mathbf{y},k,t_k}{\mathbf{b}}_{\mathbf{y},k,t_k}^T\right)\mathbf{W}{\mathbf{v}}_k{\mathbf{v}}_k^T+{\mathbf{b}}_{\mathbf{x},k,t_k}^T\mathbf{W}{\mathbf{v}}_k{\mathbf{b}}_{\mathbf{y},k,t_k}^T\right]\mathbf{W}{\mathbf{v}}_k \\ -2t_k^{-3}{\tilde{\mathcal{P}}}_{k,t_k}^2{\mathbf{b}}_{\mathbf{x},k,t_k}^T\mathbf{W}{\mathbf{v}}_k{\mathbf{b}}_{\mathbf{y},k,t_k}^T\mathbf{W}{\mathbf{v}}_k{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k+O_p\left\{{\alpha }_n{c}_k{t}_k^{-2}\right\}. \end{gathered}$$ (106)

Therefore, we have characterized the terms involving $t_k^{-1}$ for the desired first order asymptotic expansion. That is, by (106) we have

$$\begin{gathered} {\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}={\tilde{\mathcal{P}}}_{k,t_k}{A}_{\mathbf{x},k,t_k}{A}_{\mathbf{y},k,t_k}-t_k^{-1}{A}_{\mathbf{x},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}({\mathbf{b}}_{\mathbf{y},k,t_k}^T+A_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k \\ -t_k^{-1}{A}_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}({\mathbf{b}}_{\mathbf{x},k,t_k}^T+A_{\mathbf{x},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k+O_p\left\{\left({\alpha }_n{c}_k+1\right)t_k^{-2}\right\}. \end{gathered}$$ (107)

Thus if ${\sigma }_k^2=t_k^{-2}{\tilde{\mathcal{P}}}_{k,t_k}^2\mathbb{E}{\left[(A_{\mathbf{x},k,t_k}{\mathbf{b}}_{\mathbf{y},k,t_k}^T+A_{\mathbf{y},k,t_k}{\mathbf{b}}_{\mathbf{x},k,t_k}^T+2A_{\mathbf{x},k,t_k}{A}_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k\right]}^2\gg {\left({\alpha }_n{c}_k+1\right)}^2{t}_k^{-4}~{\sigma }_n^2{\left(\left|A_{\mathbf{x},k,t_k}\right|+\left|A_{\mathbf{y},k,t_k}\right|\right)}^2{t}_k^{-4}+t_k^{-4}{\left({\alpha }_n{c}_k+1\right)}^2{t}_k^{-4}~{\sigma }_n^2\left(\left|A_{\mathbf{x},k,t_k}\right|+\left|A_{\mathbf{y},k,t_k}\right|\right){\left({\alpha }_n{c}_k+1\right)}^2{t}_k^{-4}~{\sigma }_n^2{\left(\left|A_{\mathbf{x},k,t_k}\right|+\left|A_{\mathbf{y},k,t_k}\right|\right)}^2{t}_k^{-4}+t_k^{-4}{}^2{t}_k^{-4}+t_k^{-4}2A_{\mathbf{x},k,t_k}{A}_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T,{\mathbf{v}}_k)$ and $(A_{\mathbf{x},k,t_k}{\mathbf{b}}_{\mathbf{y},k,t_k}^T+A_{\mathbf{y},k,t_k}{\mathbf{b}}_{\mathbf{x},k,t_k}^T+2A_{\mathbf{x},k,t_k}{A}_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T,{\mathbf{v}}_k)$ is W¹-CLT, then (33) holds, where ~ means the asymptotic order. This concludes the proof of Theorem 4.

A.6. Proof of Theorem 5

We have characterized the first order asymptotic expansion for the bilinear form ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}$ in the proof of Theorem 4 in Section A.5, where x and y are two arbitrary n-dimensional unit vectors. We now proceed with investigating the higher order (which is second order) asymptotic expansion for the same bilinear form. More specifically, the proof of Theorem 5 involves further expansion for the $O_p\left\{{\alpha }_n{c}_k{t}_k^{-2}\right\}$ term given in (106).

To gain some intuition, let us recall (75) and compare with (77)–(81). By Lemma 4, we see that the order O_p(α_n|z|⁻³) comes from the terms of form ${\mathbf{x}}^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right)\mathbf{y}/z^3$. Therefore, to obtain a higher order expansion we need to identify all terms of form ${\mathbf{x}}^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right)\mathbf{y}/z^3$. It follows from (75) and Lemmas 4 and 5 that

$$\begin{gathered} {\mathbf{x}}^T\mathbf{G}(z)\mathbf{y}=-z^{-1}{\mathbf{x}}^T\mathbf{y}-z^{-2}{\mathbf{x}}^T\mathbf{W}y-\frac{{\mathbf{x}}^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right)\mathbf{y}}{z^3} \\ -\sum _{l=2}^L{z}^{-(l+1)}{\mathbf{x}}^T\mathbb{E}{\mathbf{W}}^l\mathbf{y}+O_p\left(|z{|}^{-4}+{\alpha }_n^2|z{|}^{-4}\right). \end{gathered}$$ (108)

Moreover, using similar arguments as for proving (101) and (103) but expanding to higher orders we can obtain

$$\begin{gathered} {\left\{{\widehat{t}}_k^2{\mathbf{v}}_k^T\left[{\mathbf{G}}^{\prime }\left({\widehat{t}}_k\right)-{\mathbf{F}}_k^{\prime }\left({\widehat{t}}_k\right)\right]{\mathbf{v}}_k\right\}}^{-1}={\tilde{\mathcal{P}}}_{k,t_k}\{1-2t_k^{-1}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k-t_k^{-2}{\tilde{\mathcal{P}}}_{k,t_k} \\ \times \left[3{\mathbf{v}}_k^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k-2{\left({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\right)}^2\right]\}+O_p\left({\alpha }_n^2{\left|t_k\right|}^{-3}\right) \end{gathered}$$ (109)

and

$$\begin{gathered} {\widehat{t}}_k{\mathbf{u}}^T\left[\mathbf{G}\left({\widehat{t}}_k\right)-{\mathbf{F}}_k\left({\widehat{t}}_k\right)\right]{\mathbf{v}}_k=A_{\mathbf{u},k,t_k}-t_k^{-1}{\mathbf{u}}^T\mathbf{W}{\mathbf{v}}_k \\ +t_k^{-1}\mathcal{R}\left(\mathbf{u},{\mathbf{V}}_{-k},t_k\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\mathbf{W}{\mathbf{v}}_k+t_k^{-2}{\mathbf{u}}^T\mathbf{W}{\mathbf{v}}_k{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k \\ -t_k^{-2}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\mathcal{R}\left(\mathbf{u},{\mathbf{V}}_{-k},t_k\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\mathbf{W}{\mathbf{v}}_k \\ +t_k^{-2}\mathcal{R}\left(\mathbf{u},{\mathbf{V}}_{-k},t_k\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k \\ -t_k^{-2}{\mathbf{u}}^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k+2t_k^{-3}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\mathcal{R}\left(\mathbf{u},{\mathbf{V}}_{-k},t_k\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-2} \\ \times {\mathbf{V}}_{-k}^T\mathbf{W}{\mathbf{v}}_k+O_p\left({\alpha }_n^2{\left|t_k\right|}^{-3}\right), \end{gathered}$$ (110)

where u represents both x and y as mentioned before.

Using the representations (100) and (102), and by the asymptotic expansions (109)–(110), we can obtain the $O_p\left(t_k^{-2}\right)$ term for the desired second order asymptotic expansion as follows

$$\begin{gathered} {\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}=\frac{{\widehat{t}}_k^2{\mathbf{x}}^T\left[\mathbf{G}\left({\widehat{t}}_k\right)-{\mathbf{F}}_k\left({\widehat{t}}_k\right)\right]{\mathbf{v}}_k{\mathbf{v}}_k^T\left[\mathbf{G}\left({\widehat{t}}_k\right)-{\mathbf{F}}_k\left({\widehat{t}}_k\right)\right]\mathbf{y}}{{\widehat{t}}_k^2{\mathbf{v}}_k^T\left[{\mathbf{G}}^{\prime }\left({\widehat{t}}_k\right)-{\mathbf{F}}_k^{\prime }\left({\widehat{t}}_k\right)\right]{\mathbf{v}}_k} \\ =({\tilde{\mathcal{P}}}_{k,t_k}\times \left\{1-2t_k^{-1}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k-t_k^{-2}{\tilde{\mathcal{P}}}_{k,t_k}\left[3{\mathbf{v}}_k^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k-2{\left({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\right)}^2\right]\right\} \\ +O_p\left({\alpha }_n^2{\left|t_k\right|}^{-3}\right))\left[{\widehat{t}}_k{\mathbf{x}}^T\left[\mathbf{G}\left({\widehat{t}}_k\right)-{\mathbf{F}}_k\left({\widehat{t}}_k\right)\right]{\mathbf{v}}_k\right]\left[{\widehat{t}}_k{\mathbf{v}}_k^T\left[\mathbf{G}\left({\widehat{t}}_k\right)-{\mathbf{F}}_k\left({\widehat{t}}_k\right)\right]\mathbf{y}\right] \\ =-A_{\mathbf{x},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-1}({\mathbf{b}}_{\mathbf{y},k,t_k}^T+A_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k-A_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-1} \\ \times ({\mathbf{b}}_{\mathbf{x},k,t_k}^T+A_{\mathbf{x},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k \\ +{\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-2}\left[2{\tilde{\mathcal{P}}}_{k,t_k}\left(A_{\mathbf{x},k,t_k}{\mathbf{b}}_{\mathbf{x},k,t_k}^T+A_{\mathbf{y},k,t_k}{\mathbf{b}}_{\mathbf{y},k,t_k}^T\right)\mathbf{W}{\mathbf{v}}_k{\mathbf{v}}_k^T+{\mathbf{b}}_{\mathbf{x},k,t_k}^T\mathbf{W}{\mathbf{v}}_k{\mathbf{b}}_{\mathbf{y},k,t_k}^T\right]\mathbf{W}{\mathbf{v}}_k \\ +2A_{\mathbf{x},k,t_k}{A}_{\mathbf{y},k,t_k}{\left({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\right)}^2+A_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}\{t_k^{-2}{\mathbf{x}}^T\mathbf{W}{\mathbf{v}}_k{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k-t_k^{-2}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\mathcal{R}\left(\mathbf{x},{\mathbf{V}}_{-k},t\right) \\ \times {\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\mathbf{W}{\mathbf{v}}_k\} \\ +A_{\mathbf{x},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}\{t_k^{-2}{\mathbf{y}}^T\mathbf{W}{\mathbf{v}}_k{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k-t_k^{-2}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\mathcal{R}\left(\mathbf{y},{\mathbf{V}}_{-k},t\right) \\ \times {\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\mathbf{W}{\mathbf{v}}_k\} \\ +A_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-2}\mathcal{R}\left(\mathbf{x},{\mathbf{V}}_{-k},t_k\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k \\ +A_{\mathbf{x},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-2}\mathcal{R}\left(\mathbf{y},{\mathbf{V}}_{-k},t_k\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k \\ -{\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-2}\left(A_{\mathbf{y},k,t_k}{\mathbf{x}}^T+A_{\mathbf{x},k,t_k}{\mathbf{y}}^T\right)\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k \\ -3t_k^{-2}{A}_{\mathbf{x},k,t_k}{A}_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k+O_p\left\{\left({\alpha }_n^2{c}_k+1\right){\left|t_k\right|}^{-3}\right\}. \end{gathered}$$ (111)

In contrast to the small order term $O_p\left\{{\alpha }_n{c}_k{t}_k^{-2}\right\}$ in (106) from the first order asymptotic expansion, we now have the small order term $O_p\left\{\left({\alpha }_n^2{c}_k+1\right){\left|t_k\right|}^{-3}\right\}$ from the second order asymptotic expansion.

Let us conduct some simplifications for the expressions given in the above asymptotic expansions in (111). A combination of (106) and (111) shows that the asymptotic distribution is determined by

$$\begin{gathered} -A_{\mathbf{x},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-1}({\mathbf{b}}_{\mathbf{y},k,t_k}^T+A_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k \\ -A_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-1}({\mathbf{b}}_{\mathbf{x},k,t_k}^T+A_{\mathbf{x},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T)\mathbf{W}{\mathbf{v}}_k \\ +{\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-2}\left[2{\tilde{\mathcal{P}}}_{k,t_k}\left(A_{\mathbf{x},k,t_k}{\mathbf{b}}_{\mathbf{x},k,t_k}^T+A_{\mathbf{y},k,t_k}{\mathbf{b}}_{\mathbf{y},k,t_k}^T\right)\mathbf{W}{\mathbf{v}}_k{\mathbf{v}}_k^T+{\mathbf{b}}_{\mathbf{x},k,t_k}^T\mathbf{W}{\mathbf{v}}_k{\mathbf{b}}_{\mathbf{y},k,t_k}^T\right]\mathbf{W}{\mathbf{v}}_k \\ +2A_{\mathbf{x},k,t_k}{A}_{\mathbf{y},k,t_k}{\left({\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\right)}^2+A_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}\{t_k^{-2}{\mathbf{x}}^T\mathbf{W}{\mathbf{v}}_k{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k-t_k^{-2}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\mathcal{R}\left(\mathbf{x},{\mathbf{V}}_{-k},t\right) \\ \times {\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\mathbf{W}{\mathbf{v}}_k\} \\ +A_{\mathbf{x},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}\{t_k^{-2}{\mathbf{y}}^T\mathbf{W}{\mathbf{v}}_k{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k-t_k^{-2}{\mathbf{v}}_k^T\mathbf{W}{\mathbf{v}}_k\mathcal{R}\left(\mathbf{y},{\mathbf{V}}_{-k},t\right) \\ \times {\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\mathbf{W}{\mathbf{v}}_k\} \\ +A_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-2}\mathcal{R}\left(\mathbf{x},{\mathbf{V}}_{-k},t\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k \\ +A_{\mathbf{x},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-2}\mathcal{R}\left(\mathbf{y},{\mathbf{V}}_{-k},t\right){\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k \\ -{\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-2}\left(A_{\mathbf{y},k,t_k}{\mathbf{x}}^T+A_{\mathbf{x},k,t_k}{\mathbf{y}}^T\right)\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k \\ -3t_k^{-2}{A}_{\mathbf{x},k,t_k}{A}_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{v}}_k. \end{gathered}$$ (112)

To further simplify the notation, we define three terms

$${\mathbf{J}}_{\mathbf{x},\mathbf{y},k,t_k}=-{\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-1}{\mathbf{v}}_k(A_{\mathbf{y},k,t_k}{\mathbf{b}}_{\mathbf{x},k,t_k}^T+A_{\mathbf{x},k,t_k}{\mathbf{b}}_{\mathbf{y},k,t_k}^T+2A_{\mathbf{x},k,t_k}{A}_{\mathbf{y},k,t_k}{\tilde{\mathcal{P}}}_{k,t_k}{\mathbf{v}}_k^T),$$ (113)

$$\begin{gathered} {\mathbf{L}}_{\mathbf{x},\mathbf{y},k,t_k}={\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-2}{\mathbf{v}}_k\{\left[A_{\mathbf{y},k,t_k}\mathcal{R}\left(\mathbf{x},{\mathbf{V}}_{-k},t\right)+A_{\mathbf{x},k,t_k}\mathcal{R}\left(\mathbf{y},{\mathbf{V}}_{-k},t\right)\right] \\ \times {\left[{\mathbf{D}}_{-k}^{-1}+\mathcal{R}\left({\mathbf{V}}_{-k},{\mathbf{V}}_{-k},t_k\right)\right]}^{-1}{\mathbf{V}}_{-k}^T+A_{\mathbf{y},k,t_k}{\mathbf{x}}^T+A_{\mathbf{x},k,t_k}{\mathbf{y}}^T \\ +3A_{\mathbf{x},k,t_k}{A}_{\mathbf{y},k,t_k}{\mathbf{v}}_k^T\}, \end{gathered}$$ (114)

$$\begin{gathered} {\mathbf{Q}}_{\mathbf{x},\mathbf{y},k,t_k}={\mathbf{L}}_{\mathbf{x},\mathbf{y},k,t_k}-{\tilde{\mathcal{P}}}_{k,t_k}{t}_k^{-2}{A}_{\mathbf{x},k,t_k}{A}_{\mathbf{y},k,t_k}{\mathbf{v}}_k{\mathbf{v}}_k^T \\ +2{\tilde{\mathcal{P}}}_{k,t_k}^2{t}_k^{-2}{\mathbf{v}}_k\left(A_{\mathbf{x},k,t_k}{\mathbf{b}}_{\mathbf{x},k,t_k}^T+A_{\mathbf{y},k,t_k}{\mathbf{b}}_{\mathbf{y},k,t_k}^T\right). \end{gathered}$$ (115)

Note that all the three matrices defined in (113)–(115) are of rank one and the identity x^T Ay = tr(Ayx^T ) holds for any matrix A and vectors x and y. Thus in view of (113)–(115), the lengthy expression given in (112) can be rewritten in a compact form as

$$\mathrm{tr}\left[\mathbf{W}{\mathbf{J}}_{\mathbf{x},\mathbf{y},k,t_k}-\left({\mathbf{W}}^2-\mathbb{E}{\mathbf{W}}^2\right){\mathbf{L}}_{\mathbf{x},\mathbf{y},k,t_k}\right]+\mathrm{tr}(\mathbf{W}{\mathbf{v}}_k{\mathbf{v}}_k^T)\mathrm{tr}\left(\mathbf{W}{\mathbf{Q}}_{\mathbf{x},\mathbf{y},k,t_k}\right).$$ (116)

So far we have shown that the second order expansion of ${\mathbf{x}}^T{\widehat{\mathbf{v}}}_k{\widehat{\mathbf{v}}}_k^T\mathbf{y}$ is given in (111). Note that ${\tilde{\sigma }}_k^2$ defined in (32) is essentially the variance of (116). Thus if ${\tilde{\sigma }}_k^2\gg {\left({\alpha }_n^2{c}_k+1\right)}^2{t}_k^{-6}~{\sigma }_n^4{\left(\left|A_{\mathbf{x},k,t_k}\right|+\left|A_{\mathbf{y},k,t_k}\right|\right)}^2{t}_k^{-6}+t_k^{-6}$, then (116) is the leading term of (111). Furthermore, the assumption of ${\sigma }_k^2=O\left({\tilde{\sigma }}_k^2\right)$ entails that the first order expansion in Theorem 4 does not dominate the second order expansion. Therefore, we see that the asymptotic distribution in Theorem 5 is determined by the joint distribution of the three random variables specified in expression (116). This completes the proof of Theorem 5.