On the Multivariate Extremal Index

Abstract

The exceedance point process approach of Hsing et al. is extended to multivariate stationary sequences and some weak convergence results are obtained. It is well known that under general mixing assumptions, high level exceedances typically have a limiting Compound Poisson structure where multiple events are caused by the clustering of exceedances. In this paper we explore (a) the precise effect of such clustering on the limit, and (b) the relationship between point process convergence and the limiting behavior of maxima. Following this, the notion of multivariate extremal index is introduced which is shown to have properties analogous to its univariate counterpart. Two examples of bivariate moving average sequences are presented for which the extremal index is calculated in some special cases.

1. Introduction

Extreme value theory for multivariate iid sequences has been studied for quite some time now but attention to the dependent case has been relatively recent. For univariate sequences it is known that local dependence causes extreme values to occur in clusters, which in turn results in a stochastically smaller distribution for the maximum than if the observations were independent. We begin with a brief review of these results, which we shall later extend to the multivariate case.

Let {ξ_n} be a univariate stationary sequence. Write M_n = max{ξ₁,…,ξ_n} and for τ>0, let {u_n(τ)} denote a sequence satisfying lim_n→∞np{ξ₁>u_n(τ)}=τ. Under quite general mixing assumptions there exist constants 0≤ θ′≤ θ″≤1 such that

$$\begin{array}{c}\underset{n\to \infty }{\lim \sup }P\left\{M_n\le u_n\left(\tau \right)\right\}={\mathrm{e}}^{\theta \tau }\text{and}\hfill \\ \underset{n\to \infty }{\lim \inf }P\left\{M_n\le u_n\left(\tau \right)\right\}={\mathrm{e}}^{{\theta }^{″}\tau }\hfill \end{array}$$

for all τ. (See Ref. [1], although the idea actually dates back to Refs. [2–4].) Thus if P{M_n≤u_n(τ₀)} converges for some τ₀, then θ′=θ″(=θ, say) and hence lim_n→∞ P{M_n≤u_n (τ)}=c^−θr for all τ>0. The common value θ is then called the extremal index of {ξ_n}. We shall assume θ to be positive whenever it exists, since the case θ=0 corresponds to a degenerate limiting distribution for M_n. Note that θ=1 for iid sequences. Let $\left\{{\widehat{\xi }}_n\right\}$ be an iid sequence with ${\widehat{\xi }}_1{=}^d{\xi }_1$, called the associated iid sequence, and write ${\widehat{M}}_n=\max \left\{{\widehat{\xi }}_1,\dots ,{\widehat{\xi }}_n\right\}$. If {ξ_n} has extremal index θ and lim_n→∞ P{M_n≤v_n(t)} = H(t) for a suitable family of normalizing constants {v_n(t)}, then it follows (upon identifying e^−θr with H(t)) that ${\lim }_{n\to \infty }P\left\{{\widehat{M}}_n\le v_n\left(t\right)\right\}=\widehat{H}\left(t\right)$where

$$H\left(t\right)=\widehat{H}{\left(t\right)}^{\theta }.$$ (1)

The extremal index is thus a measure of the effect of dependence on the limiting distribution of M_n. The stochastically smaller limiting distribution of M_n is in fact a direct result of the clustering of extremes, as explained below. See Ref. [5] for details.

For fixed τ>0 let the exceedance point process $N_n=N_n^{\left(r\right)}$ be defined by

$$\begin{array}{ll}{N}_n\left(B\right)={\displaystyle \sum _{i=1}^n{l}_{\left({\xi }_n>n_n\left(\tau \right)\right)}{l}_{\left\{ln\in B\right\}},}\hfill & B\subset \left[0,1\right],\hfill \end{array}$$

where I_A denotes the indicator function of the event A. Then for a broad class of weakly dependent sequences, the limit in distribution of N_n, if it exists, is a Compound Poisson process with intensity θτ and multiplicity distribution π on {1,2,…}. The Poisson events may in fact be regarded as the positions of “cxceedance clusters” while the multiplicities correspond to cluster sizes. More explicitly, one may divide the n observations into k_n blocks of roughly equal size and regard exceedances within each block as forming a single “cluster”, so that the cluster sizes are given by N_n(J_i), i=l,…,k_n,where $J_{\mathrm{r}}\left(=J_{n,t}\right)=\left(\frac{i-1}{k_n},\frac{i}{k_n}\right]$. For a suitable choice of k_n depending on the mixing rate of {ξ_n}, one then has

$$\begin{array}{ll}\underset{n\to \infty }{\lim }P\left\{N_n\left(J_1\right)=j|N_n\left(J_1\right)>0\right\}=n\left(j\right).\hfill & j\ge 1.\hfill \end{array}$$

and

$$\begin{array}{c}\underset{n\to \infty }{\lim }{P}^{k_n}\left\{N_n\left(J_1\right)=0\right\}=\underset{n\to \infty }{\lim }P\left\{N_n\left[0,1\right]=0\right\}\\ =\underset{n\to \infty }{\lim }P\left\{M_n\le u_n\left(\tau \right)\right\}={\mathrm{c}}^{\theta \tau },\end{array}$$

so that in particular, lim_n→∞ k_n P{N_n(J_t)>0}=θτ. Hence

$$\begin{array}{c}\underset{n\to \infty }{\lim }E{N}_n\left[0,1\right]=\underset{n\to \infty }{\lim }{k}_{n}F{N}_n\left(J_1\right)\\ \underset{n\to \infty }{\lim }{k}_{n}E\left(N_n\left(J_1\right)|N_n\left(J_1\right)>0)P\left\{N_n\left(J_1\right)>0\right\}\right)\\ =\theta \tau \underset{n\to \infty }{\lim }E\left(N_n\left(J_1\right)|N_n\left(J_1\right)>0\right),\end{array}$$

while on the other hand, lint_n→∞ EN_n[0,1]=lim_n→∞ nP{ξ₁>u_n(τ)}=τ. The cluster size distribution and the extremal index are therefore related by

$$\underset{n\to \infty }{\lim }E\left(N_n\left(J_1\right)|N_n\left(J_1\right)>0\right)=1/\theta .$$ (2)

Now let {ξ_n=(ξ_nl,…,ξ_nd), n∈ℤ} be a multivariate stationary sequence where d≥1 is a fixed integer, and write M_n=(M_nl,…, M_nd) where M_nj = max{ξ_ni,…,ξ_nj}, j−1,…,d. The study of multivariate extremes began in the early 1950s, focusing mainly on the limiting behavior of M_n under a linear normalization, when the observations are iid. The resulting class of limiting distributions was characterized in Ref. [6] and domains of attraction criteria were given in Ref. [7]. See also Ref. [8]. Chapter 5, for an account of the literature surrounding this theory. For stationary sequences satisfying a general mixing assumption, it is known (see Refs. [9, 10], and Theorem 1.1 below) that the class of limiting distributions of M_n is the same as for iid sequences. In this paper we explore the precise effect of dependence on the limiting distribution by extending the univariate theory described above to the multivariate case. Essentially, this involves studying the inter-relationship between the two dependence structures present, one due to dependence over time and the other due to the dependence between the various components of the multivariate observations. The ideas become most transparent when presented in terms of so-called dependence functions [8]. Here we adopt the slightly modified definition found in Ref. [9]. A distribution function D on [0,1]^d it called a dependence function if D_j(D_j{u))=D_l(u), u∈|0,1|, j=1,…,d, where the subscript j signifies they jth marginal. The dependence function of a distribution F on IR^d is defined by D_f(u)=P{F₁(X₁)≤ u_l,…,F_d(X_d)≤u_d}, u=(u_l,…,u_d)∈ [0,1]^d, where (X₁,…,X_d) is a random vector with distribution F. More generally, any dependence function satisfying F(x)=D(F₁(x₁),…,F_d(X_d)) could be defined to be a dependence function of F, although the present choice is a natural one.

Write T=(0,1)^d\{l} where l=(l,…,1)∈IR^d, and for t=(t_l,…,t_d)∈T, let v_n(t)=(v_nl(t_l),…,v_nd(t_d)) where v_nj(t_j) satisfies lim_n→∞ nP{ξ_lj>v_nj(t_j)}=− logt_j. Let H_ndenote the distribution function of M_n(i.e., H_n(x)=P{M_n≤x}), with marginals H_nj, j=l,…,d. Then (see Refs. [8, 11]).

$$H_n\left(v_n\left(\mathbf{t}\right)\right){{\to }^{\prime }}^{\prime }H\left(\mathbf{t}\right)$$ (3)

if and only if

$$H_{nj}\left(v_{nj}\left(t_j\right)\right){\to }^H{H}_t\left(t_j\right),j=1,\dots ,d,\text{and}{D}_{jtn}\left(\mathbf{t}\right){\to }^{\mathrm{n}}{D}_H\left(\mathbf{t}\right).$$

The limiting behavior of M_n can therefore be separated into two parts, one pertaining to the convergence of the marginals (a univariate problem) and the other to the convergence of the dependence functions. Here we focus attention exclusively on the latter. It should be noted that the choice of normalising constants does not affect the dependence function of the limit distribution H, but only alters the marginals (sec Ref. [9], Lemma 3.2). Since our main interest is in the dependence function, the present choice of normalising constants is appropriate in view of the fact that it results in Uniform[0,1] marginals for the limit distribution when {ξ_n} is iid, so that in particular D_H=H. According to Theorem 3.3 of Ref. [9], the class of all possible limits H in Eq. (3) (for iid {ξ_n}) is precisely the class of extreme dependence functions, that is those that satisfy

$$D^n\left(\mathbf{t}\right)=D\left(t_1^n,\dots ,t_d^n\right)$$ (4)

for each n>1 and t=(t_l,…,t_d)∈[0,1]^d. Theorem 1.1 below shows that the same is true also if {ξ_n} is a stationary sequence satisfying the following mixing condition.

For t∈T, let

$${ℬ}_k^1\left(v_n\left(\mathbf{t}\right)\right)=\sigma \left\{\left({\xi }_{ij}>v_{nj}\left(t_j\right)\right):k\le i\le l,j=1,\dots ,d\right\}$$

and for 1≤l≤n−1 define

$$\begin{array}{c}{\alpha }_{n,l}=\sup \{\left|P\left(A\cap B\right)-P\left(A\right)P\left(B\right)\right|:A\in {ℬ}_1^k\left(v_n\left(\mathbf{t}\right)\right),\\ B\in {ℬ}_{k+l}^n\left(v_n\left(\mathbf{t}\right)\right),1\le k<k+l\le n\}.\end{array}$$

The mixing condition Δ(v_n(t}) is then said to hold if ${\alpha }_{n,I_n}\to 0$ for some sequence {l_n} satisfying l_nln→0. This is the multivariate version of the mixing condition used in Ref. [5] and is slightly stronger than the D(u_n) condition in Ref. [9]. Henceforth {ξ_n} will be assumed to satisfy Δ(v_n(t)), for some or all t, as required.

Theorem 1.1. Let {ξ_n} satisfy Δ(v_n(t)) for all t∈T and suppose that P{M_n≤v_n(t))→^H H(t), non-degenerate. Then D_n is an extreme dependence function and hence, in particular, H(t^c)=H^c(t) for each t∈[0.1]^d and c>0 (where ${\mathbf{t}}^c=\left(t_1^c,\dots ,t_d^c\right)$).

Proof: The first part is an immediate consequence of Theorem 4.2 of Ref. [9] while the second part follows from the definition of extreme dependence functions upon noting that (by the univariate theory described above), the marginals of H are of the form $H_j\left(t_j\right)=t_j^{{\theta }_j}$where θ_j is the extremal index of {ξ_nj}, the jth-component sequence of {ξ_n}.

In the next section we apply the exceedance point process approach to multivariate extremes and obtain some weak convergence results. The multivariate extremal index is then defined (in Sec. 3), based on the multivariate analogue of Eq. (1). It is seen to be a function of only d − 1 variables and its properties naturally extend those of the univariate extremal index. Finally in Sec. 4 we consider two examples of bivariate moving average sequences for which the computation of the extremal index is demonstrated.

2. Exceedance Point Processes

Fix t∈T and let ${\delta }_{ij}=I_{\left\{{\xi }_{ij}>{\nu }_{nj}\left({\tau }_j\right)\right\},}i=1,\dots ,n,j=1,\dots d$, and put δ_i=(δ_il,…,δ_id). The multivariate exceedance point process $N_n=N_n^{\left(\mathbf{t}\right)}$ is then defined by

$$\begin{array}{cc}{N}_n\left(B\right)={\displaystyle \sum _{i=1}^n{I}_{\left(itn\in B\right)}{\delta }_1,}& B\in \left[0,1\right].\end{array}$$ (5)

Assume that {ξ_n} satisfies Δ(v_n(t)). If also N_n→^d N₀, then it may be shown (as in the univariate case) that the limit N₀ is a point process on [0,1] which is of Compound Poisson type. More precisely, the Laplace Transform of N₀ is given by

$$\begin{array}{c}-\log E\exp \{-{\displaystyle \sum _{j=1}^d{\displaystyle {\int }_{\left|0,1\right|}{f}_j\mathrm{d}{N}_{0j}}}\}=v{\displaystyle {\int }_{x\in \left|0,1\right|}{\displaystyle {\int }_{y\in x_4^{d+}} }}\\ (1-\exp \{-{\displaystyle \sum _{j=1}^{d}y{f}_j\left(x\right)}\})\mathrm{d}\pi \left(\mathrm{y}\right)\mathrm{d}x.\end{array}$$ (6)

Here N_0j denotes the jth-component of N_0, v is a positive constant, π is a probability distribution on ${\mathbb{Z}}_{+}^{d1}={\left\{0,1,2,\dots \right\}}^d\backslash \left\{0\right\}$ and f_j’s are non-negative functions on [0,1].

Let {k_n} be any sequence of positive integers satisfying

$$\begin{array}{cc}{k}_n\to \infty ,k_n{l}_n/n\to 0,\text{and}{k}_n{\alpha }_{n{\lambda }_n}\to 0,& \text{as}n\to \infty .\end{array}$$ (7)

Set r_n=[n/k_n] (the largest integer not exceeding n/k_n) and put J_ni=[0,r_n/n]. Define the probability distribution π, on ${\mathbb{Z}}_{+}^{d_1}$ by

$${\pi }_{\pi }\left(\mathrm{y}\right)=P\left\{N_n\left(J_{\pi 1}\right)=\mathrm{y}|N_{\pi }\left(J_{n1}\right)\ne 0\right\},\mathrm{y}\in {\mathbb{Z}}_{+}^{dt}.$$

The following theorem which gives a useful characterization of the convergence of N_n is an immediate consequence of the results in Sec. 5 of Ref. [12].

Theorem 2.1. N_n→^d N_o if and only if π_n→^w π and P {M_n≤v_n(t)}→ e^−v, and in that case the Laplace Transform of N₀ is given by Eq. (6).

Next we consider the iid case in some detail and obtain an interesting connection with Theorem 5.3.1 of Ref. [8].

Proposition 2.2. Let {ξ_n} be iid and for fixed t∈T let N_n be defined by Eq. (5). If N_n→^d N₀ then the multiplicity distribution π in Eq. (6) is supported on the set S−{0,1}^d\{0}.

Proof: Observe that Δ(v_n(t)) is trivially satisfied since α_n.1=0 so that we may take l_n≡1 and k_n=n. Then ${\pi }_{\pi }\left(\mathrm{y}\right)=P\left\{{\delta }_1=\mathrm{y}|{\xi }_1\cancel{\le }{v}_{\kappa }\left(\mathbf{t}\right)\right\},\mathrm{y}\in {\mathbb{Z}}_{+}^{d1}$, which is clearly supported on S. The result is now immediate since S is a closed set and π_n→^H π by Theorem 2.1.

Making the dependence on t explicit, we now write $N_n=N_n^{\left(t\right)}$, $N_0=N_0^{\left(t\right)}$, v=v^(t). In addition we shall require the following notation from Ref. [8]. For 1≤k≤d, let j(k)=(j₁,…,j_k) denote a vector with integer-valued components 1≤j₁<j₂<…<j_k≤d, and for x=(x₁,…,x_d)∈IR^d write x_j(k)=(x_j1,…,x_jk). Define the “survival function”

$$G\left(\mathrm{x}\right)=P\left\{{\xi }_{11}>x_1,\dots ,{\xi }_{1d}>x_d\right\}$$

and write $G_{\mathrm{j}\left(k\right)}\left(\mathbf{x}\right)=P\left\{{\xi }_{1j_1}>x_{j_1},\dots ,{\xi }_{1j_k}>x_{j_{\mathrm{k}}}\right\}$. For each j(k), let y_j(k) denote the element in S={0,1 }^d\{0} whose jth component equals 1 if and only if j=j_i for some i=1,…,k. (This defines a natural 1-1 correspondence between S and the j(k)’s.)

Theorem 2.3. Let {ξ_n} be iid. Then $N_0^{\left(\mathbf{t}\right)}{\to }^d{N}_0^{\left(l\right)}$ for some fixed t∈T if and only if

$$\underset{n\to \infty }{\lim }n{G}_{\mathrm{j}\left(k\right)}\left(v_n\left(\mathbf{t}\right)\right)=h_{\mathrm{j}\left(k\right)}\left(\mathbf{t}\right)<\infty$$ (8)

for each j(k).1≤k≤d. In that case $N_0^{\left(\mathbf{t}\right)}$ has Laplace Transform given by Eq. (6) with

$$v^{\left(\mathbf{t}\right)}={\displaystyle \sum _{k=1}^d{\left(-1\right)}^{k+1}}{\displaystyle \sum _{1\le j<,\dots <j_k\le d}{h}_{\mathrm{j}\left(k\right)}\left(\mathbf{t}\right)}$$ (9)

and with π^(t) determined by the relations

$$h_{\mathrm{j}\left(k\right)}\left(\mathbf{t}\right)=v^{\left(\mathbf{t}\right)}{\displaystyle \sum _{y\ge y_{\mathrm{j}\left(k\right)}}{\pi }^{\left(\mathbf{t}\right)}\left(\mathrm{y}\right).}$$ (10)

Proof: Write $S_k\left(\mathbf{t}\right)={\displaystyle {\sum }_{1\le j1<\dots <j_1\le d}{G}_{j\left(k\right)}\left(\mathbf{t}\right)}$, so that $P\left\{{\xi }_1\cancel{\le }{v}_n\left(\mathbf{t}\right)\right\}={\displaystyle {\sum }_{k=1}^d{\left(-1\right)}^{k+1}{S}_k\left(v_n\left(\mathbf{t}\right)\right)}$. If Eq. (8) holds for each k, then

$$\underset{n\to \infty }{\lim }nP\left\{{\xi }_1\cancel{\le }{v}_n\left(\mathbf{t}\right)\right\}={\displaystyle \sum _{k=1}^d{\left(-1\right)}^{k+1}}{\displaystyle \sum _{1\le j_1<,\dots <j_k<d}{h}_{\mathrm{J}\left(k\right)}\left(\mathbf{t}\right)=v^{\left(\mathbf{t}\right)},}$$ (11)

and hence ${\lim }_{n\to \infty }P\left\{M_R\le v_n\left(\mathbf{t}\right)\right\}={\mathrm{e}}^{-v^{10}}$.

Next observe that for each j(k). 1≤k≤d,

$$G_{\mathrm{j}\left(k\right)}\left(v_{\pi }\left(\mathbf{t}\right)\right)={\displaystyle \sum _{\mathrm{y}\ge {\mathrm{y}}_{\mathrm{j}}\left(k\right)}P\left\{{\mathrm{\delta }}_1=\mathrm{y}\right\}=P\left\{{\xi }_1\cancel{\le }{v}_{\pi }\left(\mathbf{t}\right)\right\}}{\displaystyle \sum _{y\ge y_{j\left(k\right)}}{\pi }_n\left(y\right),}$$ (12)

where ${\pi }_n\left\{\mathrm{y}\right)=P\left({\delta }_1-\mathrm{y}|{\xi }_1\cancel{\le }{v}_n\left(\mathbf{t}\right)\right\}$. Moreover, this relationship is invertible in the sense that each of the probabilities π_n(y), y∈S, can be expressed as a linear combination of the G_j(k)(v_n(t))’s. Therefore by Eq. (8). lim_n→∞tπ_n(y)=π^(t)(y) (say) exists and satisfies Eq. (10). Hence by Theorem 2.1 $N_n^{\left(\mathbf{t}\right)}{\to }^d{N}_0^{\left(\mathbf{t}\right)}$ where $N_0^{\left(\mathbf{t}\right)}$ has the specified parameters. Conversely if $N_n^{\left(\mathbf{t}\right)}{\to }^d{N}_0^{\left(\mathbf{t}\right)}$ then π_n and P{M_n≤v_n(t)} converge (by Theorem 2.1 again), and hence Eq. (8) follows by virtue of Eqs. (11) and (12) □.

Corollary 2.4. Let {ξ_n} be iid. Then $N_n^{\left(\mathbf{t}\right)}{\to }^d{N}_0^{\left(\mathbf{t}\right)}$ for each t∈T if and only if P{M_n≤v_n(t)}→^w H(t). Moreover H and {v^(t),π^(t)}_t∈T determine each other.

Proof: (Sketch) The first part follows from Theorem 2.3 above and Theorem 5.3.1 of Ref. [8] which states that P{M_n≤v_n(t)}→^w H(t) if and only if Eq. (8) holds for each t∈T. Note that H(t)=e^−v(t) so that H and the v^(t)’s can be obtained from each other. Also the π^(t)’s can be obtained from the v^(t)’s by first inverting Eq. (9) to get the h_j(k)(t)’s and then inverting Eq. (10). (The inversion of Eq. (9) is carried out inductively using the fact that the weak convergence of H_n(v_n(t)) implies that of all lower dimensional marginals.) □

Analogous results for the dependent case take on a slightly different form. Let {ξ_n} be a stationary sequence satisfying Δ(v_n(t)) for each t∈T. As before let r_R=[n/k_n] where {k_n} is any sequence satisfying Eq. (7). and define

$$G_{r_{n}1\left(k\right)}\left(v_n\left(\mathbf{t}\right)\right)=P\left\{M_{r_n{j}_1}>V_{\pi j_1}\left(t_{j1}\right),\dots ,M_{r_{n}it}>v_{ji}\left(t_{ji}\right)\right\}.$$

Theorem 2.5. Let {ξ_n}be a stationary sequence satisfying Δ(v_n(t)) for each t∈T. Then P{M_R≤v_n (t))→^w H(t) if and only if

$$\underset{n\to \infty }{\lim }{k}_{\pi }{G}_{r_n\mathrm{j}\left(k\right)}\left(v_n\left(\mathbf{t}\right)\right)=h_{j\left(k\right)}\left(\mathbf{t}\right)<\infty$$

for each j(k), 1≤k≤d and t∈T, and in that case

$$H\left(\mathbf{t}\right)=\exp \{{\displaystyle \sum _{k=1}^d{\left(-1\right)}^k}{\displaystyle \sum _{1<j_1<\cdots <j_k\le d}{h}_{\mathrm{j}\left(k\right)}\left(\mathbf{t}\right)}\}.$$

Proof: Observe that the mixing condition Δ(v_n(t)) implies that {ξ_nj(k)} satisfies Δ(v_nj(k)(t)) for each j(k) (with obvious notation). Hence it may be shown as in the univariate case (see Lemma 2.1 of Ref. [1]) that

$$P\left\{M_{n\mathrm{j}\left(k\right)}\le v_{n\mathrm{j}\left(k\right)}\left(\mathbf{t}\right)\right\}-P^{k_n}\left\{M_{r_n\mathrm{j}\left(k\right)}\le v_{n\mathrm{j}\left(k\right)}\left(t\right)\right\}\to 0.$$ (13)

for each j(k). The result may therefore be proved in exactly the same way as Theorem 5.3.1 of Ref. [8]. □

Remark: Under the hypothesis of Theorem 2.5, if $N_n^{\left(\mathbf{t}\right)}{\to }^d{N}_0^{\left(\mathbf{t}\right)}$, t∈T, with parameters v^(t) and π^(t) then $P\left\{M_n\le v_n\left(\mathbf{t}\right)\right\}{\to }^{w}H\left(\mathbf{t}\right)-{\mathrm{c}}^{-{\rho }^{\left(\mathbf{t}\right)}}$, as in the iid case. However it is not possible in general to recover the π^(t)’s from H since the clustering of exceedances may cause the support of π^(t) to extend beyond S. References [9, 10] give sufficient conditions (analogous to the D¹(u_n) condition of Ref. [13]) under which clustering docs not occur, so that Corollary 2.4 can be extended to stationary sequences satisfying this condition.

A distribution function F on IR^d is said to be independent if $F\left(\mathrm{x}\right)={\prod }_{j=1}^d{F}_j\left(x_j\right),\mathbf{x}\in {\text{IR}}^d$. If {ξ_n} is iid and P{M_n≤v_n(t)}→^w H(t), then it follows from Corollary 5.3.1 of Ref. [8] that H is independent if and only if the marginals of H are pairwise independent. The analogous result for the dependent case is stated below. The proof (which is omitted) is essentially the sanie as for the iid case, but uses Theorem 2.5 instead of Theorem 5.3.1 of Ref. [8].

Corollary 2.6. Let {ξ_n} be a stationary sequence satisfying Δ(v_n(t)) for each t∈T and suppose that P {M_n≤v_n(t)}→^w H(t). Then H is independent if and only if $k_{n}P\left\{M_{r_{n}j}>v_{n}j\left(t_j\right),M_{r_n-1}>v_{nJ}\left(t_1\right)\right\}\to 0$ for each 1≤j<l≤d, t∈T, i.e., if and only if $k_n{G}_{r_n\mathrm{j}\left(k\right)}\left(v_n\left(\mathbf{t}\right)\right)\to 0$ for each j(2) and each t∈T.

It is shown in Ref. [14] that H is independent it $H\left(\mathbf{t}\right)={\displaystyle {\prod }_{j-1}^d{H}_j\left(t_j\right)}$ for some t∈(0,1)^d. Although the result in [14] only stated for iid sequences under a linear normalization, the proof essentially rests on the defining property of extreme dependence functions, namely Eq. (1). Consequently the result extends lo the present more general situation allowing dependence and non-lineal normalizations. Corollary 2.6 can therefore be improved as follows.

Corollary 2.7. Let {ξ_n} be as in Corollary 2.6 and suppose that P{M_n≤v_n(t)}→^w H(t). Then the following are equivalent:

1. H is independent,

2. $H\left(\mathbf{t}\right)={\displaystyle {\prod }_{j-1}^d{H}_j\left(t_j\right)}$for somet∈(0,1)^d,

3. $k_n{G}_{r_n\mathrm{j}\left(k\right)}\left(v_R\left(\mathbf{t}\right)\right)\to 0$for each j(2).for some t∈(0,1)^d.

It should be noted that Refs. [9, 10] give some interesting sufficient conditions for H to be independent when {ξ_n} is a stationary sequence. A natural question to ask in the present context is whether H is independent whenever Ĥ is. Proposition 3.4 gives a necessary and sufficient condition for this in terms of the extremal index, but the answer in general is negative and a counter-example can be found in [10]. It seems more plausible that the converse may be true, i.e., that Ĥ is independent whenever H is. In fact however, this too is not the case, as shown by an interesting counter-example in [15].

We conclude this section by stating a result which extends Theorem 5.1 of [5] and is proved similarly.

Theorem 2.8. Let {ξ_n} be a stationary sequence satisfying Δ(v_n(t)) for each t∈T and suppose that $N_n^{\left(\mathbf{t}\right)}{\to }^d{N}_0^{\left(\mathbf{t}\right)}$ for some t∈T. Then $N_n^{\left({\mathbf{t}}^c\right)}{\to }^d{N}_0^{\left({\tau }^d\right)}$ for each c>0 and fur thermore, $v^{\left({\mathbf{t}}^c\right)}=c{v}^{\left(\mathbf{t}\right)}$ and ${\pi }^{\left({\mathbf{t}}^c\right)}={\pi }^{\left(\mathbf{t}\right)}$ (where ${\mathbf{t}}^c=\left(t_1^c,\dots ,t_d^c\right)$).

3. The Multivariate Extremal Index

Let {ξ_n} be a stationary sequence and $\left\{{\widehat{\xi }}_n\right\}$ the associated iid sequence. Suppose that P{M_n≤v_n(t)}→^w H(t) and $P\left\{{\widehat{M}}_n\le v_n\left(\mathbf{t}\right)\right\}{\to }^w\widehat{H}\left(\mathbf{t}\right)$. The multivariate extremal index of {ξ_n} is then defined by the relation H(t)−Ĥ^θ(t)(t) (see Eq. (1)), or more explicitly

$$0\left(\mathbf{t}\right)-\log H\left(\mathbf{t}\right)/\log \widehat{H}\left(\mathbf{t}\right),\mathbf{t}\in T.$$

Obscene that θ(t) is well defined since Ĥ has Uniform[0,1] marginals and hence, 0<Ĥ(t)<1 on T. The following results describe some basic properties of the multivariate extremal index.

Proposition 3.1. Assume that {ξ_n}satisfies Δ(v_n(t)) for each t∈T and has extremal index θ(t). Then

1. θ(t)=θ(t^c) for eacht∈T and c>0, and

2. for each j=1,…,d, {ξ_nj} has extremal index θ_j=θ(t) wheret∈T has all coordinates equal to I except the jth.

(Note that by (i), θ(t) is constant along the contours L₁={t^c:c>0}, t∈T, and hence θj in (ii) is well defined.)

Proof: Recall mat (by Theorem 1.1) H(t^c)=H^c(t) and Ĥ(t^c)=Ĥ^c(t) so that (i) follows from the definition of the extremal index. Next, for t∈T with all coordinates but the jth equal to 1, P{M_n≤v_n(t)}=P{M_nj≤v_nj(t_j)} and hence

$$\underset{\pi \to \infty }{\lim }P\left\{M_{nj}\le v_{nj}\left(l_j\right)\right\}=\underset{\pi \to \infty }{\lim }P\left\{M_n\le v_n\left(\mathbf{t}\right)\right\}=H\left(\mathbf{t}\right)=H_j\left(t_j\right).$$

Therefore by Theorem 2.2 of Ref. [1], {ξ_nj} has extremal index θ_j (say) so that $H_j\left(t_j\right)=t_J^{{\theta }_j}$. Now H(t)−Ĥ^θ(t)(t) by definition of the extremal index, and for the present choice of t this is the same as $H_j\left(t_j\right)=t_j^{\theta \left(\mathbf{t}\right)}$, whence it follows that θ(t)=θ_j for all such t.

For t∈T, let ${\overline{N}}_n^{\left(\mathbf{t}\right)}$ denote the one-dimensional point process obtained from $N_{\pi }^{\left(\mathbf{t}\right)}$ via the map y→l_(y≠0) from {0,1}^d to {0,1}, i.c., ${\overline{N}}_{\pi }^{\left(\mathbf{t}\right)}\left(B\right)-{\displaystyle {\sum }_{i-1}^n{I}_{\left(i^{\prime }n\in B\right)}{I}_{\left({\delta }_i\ne \cap \right)}}$, B∈ℬ. Thus ${\widehat{N}}_n^{\left(\mathbf{t}\right)}$has unit mass at i/n if and only if ξ_i≰v_n(t). Assume that {ξ_n} satisfies Δ(v_n(t)) and with J_nl as in Sec. 2. let

$${\overline{\pi }}_n^{(\mathbf{t})}\left(y\right)=P\left\{N_n^{\left(\mathbf{t}\right)}\left(J_{n1}\right)=y|{\overline{N}}_n^{\left(\mathbf{t}\right)}\left(J_{n1}\right)>0\right\}.y\ge 1.$$

Proposition 3.2. Assume that {ξ_n}satisfies Δ(v_n(t)) for each t∈T and has extremal index θ(t). Then $\theta \left(\mathbf{t}\right)={\left({\lim }_{\alpha \to \infty }{\displaystyle {\sum }_{y>1}y{\pi }_n^{-\left(\mathbf{t}\right)}\left(y\right)}\right)}^{-1}$.

Proof: Observe that

$$\begin{array}{c}{\displaystyle \sum _{y>1}y{\pi }_n^{(\mathbf{t})}(y)}=E\left({\tilde{N}}_n^{\left(\mathbf{t}\right)}\left(J_{n1}\right)|{\tilde{N}}_n^{\left(\mathbf{t}\right)}\left(J_{n1}\right)>0\right)\frac{r_{n}P\left\{{\xi }_1\cancel{\le }{v}_n\left(\mathbf{t}\right)\right\}}{P\left\{{\tilde{N}}_n^{\left(\mathbf{t}\right)}\left(J_{n1}\right)>0\right\}}=\\ \frac{k_n{r}_n}{n}\frac{nP\left\{{\xi }_1\cancel{\le }{v}_n\left(\mathbf{t}\right)\right\}}{k_{n}P\left\{M_{r_n}\cancel{\le }{v}_n\left(\mathbf{t}\right)\right\}}.\end{array}$$

Now ${\lim }_{n\to \infty }P\left\{{\overline{M}}_n\le v_n\left(\mathbf{t}\right)\right\}=\widehat{H}\left(\mathbf{t}\right)$ and lim_n→∞ P{M_n≤v_n(t)}=H(t) (by assumption), so that lim_n→∞ nP{ξ_l≰v_n(t)}=−log Ĥ(t) and (by Eq. (13)) ${\lim }_{n\to \infty }{k}_{n}P\left\{M_{r_n}\cancel{\le }{v}_n\left(\mathbf{t}\right)\right\}=-\log H\left(\mathbf{t}\right)$. Therefore ${\lim }_{n\to \infty }{\sum }_{y\ge 1}y{\tilde{\pi }}_n^{\left(\mathbf{t}\right)}\left(y\right)=\log \widehat{H}\left(\mathbf{t}\right)/\log H\left(\mathbf{t}\right)-1/\theta \left(\mathbf{t}\right)$, as required. □

Remark: Proposition 3.2 is simply the multivariate version of Eq. (1) and shows how the extremal index is related to the clustering of “exceedances.” Indeed, according to the present viewpoint, an exceedance occurs at time i if ξ≰v_n(t), i.c., if ξ_nj>v_nj(t_j) for at least one j. Thus Propositions 3.1 and 3.2 show that while the degree of clustering may depend on t, it is constant on each L_t. Note also the connection to Theorem 2.8.

The next result gives the relation between the dependence functions of H and Ĥ, which is seen to involve the extremal index in an intricate manner.

Proposition 3.3. If {ξ_n} has extremal index θ(t), t∈T. then

$$D_H\left(t_1^{{\theta }_1},\dots ,t_d^{{\theta }_d}\right)=D_H^{\theta \left(\mathbf{t}\right)}\left(\mathbf{t}\right),\mathbf{t}\in T,$$ (14)

where θ_j is the extremal index of {ξ_n}, j=1,…,d.

Proof: By definition of the dependence function, D_H(t)=P{H_l(X_l)≤t_l,…,H_d(X_d)≤t_d} where (X_l,…, X_d) ts a random vector with distribution H. Therefore, since $H_j\left(t_j\right)=t_j^{{\theta }_j}$.

$$\begin{array}{c}{D}_H\left(t_1^{{\theta }_1},\dots ,t_d^{{\theta }_d}\right)=\left\{X_1\le t_1,\dots ,X_d\le t_d\right\}=H\left(\mathbf{t}\right)={\widehat{H}}^{\left(\theta \right)}\left(\mathbf{t}\right)\\ =D_H^{\left(\theta \right)}\left(\mathbf{t}\right),\end{array}$$

or required. □

Remarks

1. Note that s=t^c (for some c>0) if and only if log s_j/log s_d=log t_j/log t_d−a_j (say), j=1,…,d − 1. Therefore we may write L_t=L_a where a=(a₁,…,a_d−1), and hence by the remark following Proposition 3.2, θ(t)=θ(a), i.c., the extremal index is a function of d − 1 variables only.

2. By Proposition 3.3, $D_H\left(t_1^{{\theta }_1},\dots ,t_d^{{\theta }_d}\right)=D_H^{\theta \left(t\right)}\left(\mathbf{t}\right)-D_H\left({\mathbf{t}}^{\theta \left(\mathbf{t}\right)}\right)$. Also, if t∈L_a then $\left(t_1^{{\theta }_1},\dots ,t_d^{{\theta }_d}\right)\in L_{a*}$, where a*=(a₁θ₁/θ_d,…,a_a−1θ_d−1/θ_d). Thus D_H is obtained by translating the values of D_Ĥ(=Ĥ) on L_a onto $L_{a^{*}}$.

3. While the above results illustrate some of the basic properties of the multivariate extremal index, they are far from complete. For instance; it would be useful to identify the set of all “admissible” θ(•) for a given Ĥ, that is the set of all θ(•) such that D_H(•) defined by Eq. (14) is a probability distribution on [0,1]^d. It would also be of interest to study the properties of θ(•) when one or both of Ĥ and H are independent. In this context we have the following simple result.

Proposition 3.4. If Ĥ is independent, then Ĥ is independent if and only if

$$\theta \left(\mathbf{t}\right)=\sum _{j=1}^d{\theta }_j\log t_j/\sum _{j=1}^d\log t_j,forsome\mathbf{t}\in {\left(0,1\right)}^d,$$

In particular, if both Ĥ and H are independent then θ(t) is a convex combination of the θ_j’s.

Proof: If Ĥ is independent, then $H\left(\mathbf{t}\right)-\widehat{H}{\left(\mathbf{t}\right)}^{\theta \left(\mathbf{t}\right)}={\left({\prod }_{j-1}^d{I}_j\right)}^{\theta \left(\mathbf{t}\right)}$. The conclusion follows immediately from Corollary 2.7 (iii) upon taking logarithms and noting that if H is independent, then $H\left(\mathbf{t}\right)={\displaystyle {\prod }_{j-1}^d{t}_j^{\theta j}}$.

The extremal index can be given the following more general formulation. Let $\widehat{\mu }$and μ be the probability measures on (0,1)^d corresponding to Ĥ and H, respectively. Thus for instance,

$$\mu \left(A\right)=\underset{n\to \infty }{\lim }P\left\{M_n\in v_n\left(A\right)\right\}$$

where v_n(A)={v_n(s) : s∈A}, A⊂(0,1))^d. We now define $\overline{\theta }\left(A\right)$ via the relationship $\mu \left(A\right)={\widehat{\mu }}^{\theta \left(A\right)}\left(A\right)$, or more directly $\overline{\theta }\left(A\right)=\log \mu \left(A\right)/\log \widehat{\mu }\left(A\right)$, for subsets A⊂(0,1)^d such that $\widehat{\mu }\left(A\right)>0$ and μ(A)>0.

Note that $\theta \left(\mathbf{t}\right)-\overline{\theta }\left(\left(0,t_1\right)\times \dots \times \left(0,t_d\right)\right)$for t∈T. Thus if {θ(t): t∈T} is known along with either of H or Ĥ, then it is possible at least in theory to obtain $\left\{\overline{\theta }\left(A\right):A\subset {\left(0,1\right)}^d\right\}$. In practice, however, it may not be possible to obtain $\overline{\theta }\left(A\right)$in a tractable form, but frequently one is only interested in certain special sets, typically rectangles of the form ${\prod }_{j-1}^d\left(a_j,b_j\right)$, and for such sets the computation is easy.

The definition of M_n as the vector of component wise maxima actually corresponds to regarding ξ_i as an extreme observation if ξ_ij>v_nJ(t_j) for some j. More generally, one may define ξ_i to be an extreme value if ξ_i∈v_n(A) for some A⊂(0,1)^d, in which case $\overline{\theta }\left(A\right)$ has an interpretation as a measure of the clustering of such extremes. Note that the original definition of extremes corresponds to letting A=((0,t₁)×…×(0,t_d)^c. Alternately one may consider taking A−(t₁,l)×…×(t_d,l) which corresponds to defining ξ₁ as an extreme observation if ξ_ij>v_nj(t_j) for all j. Yet another choice is $A=\left\{\mathbf{t}:\mathrm{\Sigma }{t}_j^2>c\right\}$.

4. Examples

We conclude with two examples, both involving bivariate stationary sequences.

Example 4.1 Let {η_n} be an iid sequence, and put ξ_n1=η_a, and ξ_n2=max{η_n−1, η_n}. Let F denote the distribution of ξ_n=(ξ_n1, ξ_n2) with marginals F₁ and F₂. Then $F_2\left(x\right)=P\left\{{\xi }_{n2}\le x\right\}=P\left\{{\eta }_{n-1}\le x,{\eta }_n\le x\right\}=F_1^2\left(x\right)$ and

$$F\left(x_1,x_2\right)=P\left\{{\xi }_{n1}\le x_1,{\xi }_{n2}\le x_2\right\}=\{\begin{array}{l}\begin{array}{cc}{F}_1^2\left(x_2\right),& \text{if}{x}_1\ge x_2\end{array}\\ \begin{array}{cc}{F}_1\left(x_1\right)F_1\left(x_2\right),& \text{if}{x}_1<x_{2.}\end{array}\end{array}$$

If v_nj(t_j) satisfies $F_j^n\left(v_{nj}\left(t_j\right)\right)\to t_j,j=1,2$ then ${\lim }_{n\to \infty }{F}_1^n\left(v_{n2}\left(t_2\right)\right)=t_2^{1/2}$ so that $v_{n2}\left(t_2\right)=v_{n1}\left(t_2^{1/2}\right)$. Moreover v_n1(t₁)≥v_n2(t₂) if and only if $t_1\ge t_2^{1/2}$, and so

$$H_n\left(v_n\left(t\right)\right)=P\left\{M_n\le v_n\left(t\right)\right\}{\to }^{n}H\left(t\right)=\{\begin{array}{ll}{t}_2^{1/2},\hfill & \text{if}{t}_1\ge t_2^{1/2}\hfill \\ t_{1,}\hfill & \text{if}{t}_1<t_2^{1/2}.\hfill \end{array}$$

The marginals of H are therefore H₁(t₁)=t₁ and $H_2\left(t_2\right)=t_2^{1/2}$ so that θ₁=1 and θ₂=1/2, and the dependence function of H is $D_H\left(t\right)=H\left(t_1,t_2^2\right)=t_1^t_2$. For the associated iid sequence $\left\{{\widehat{\xi }}_n\right\}$ on the other hand, it is easily verified that

$${\widehat{H}}_n\left(v_n\left(t\right)\right)=P\left\{{\widehat{M}}_n\le v_n\left(r\right)\right\}\to \widehat{H}\left(t\right)=\{\begin{array}{ll}{t}_2,\hfill & \text{if}{r}_1\ge t_2^{1/2}\hfill \\ r_1{t}_2^{1/2},\hfill & \text{if}{t}_1<t_2^{1/2}.\hfill \end{array}$$

from which it follows that

$$\theta \left(t\right)=\{\begin{array}{ll}\frac{1}{2t}\hfill & \text{if}{t}_1\ge t_2^{1/2}\hfill \\ \frac{\log t_1}{\log t_1{t}_2^1}\hfill & \text{if}{t}_1<t_2^{1/2}.\hfill \end{array}$$

We next consider a moving average sequence studied in Ref. [16].

Example 4.2, Let {Z_k=(Z_k1,Z_k2)¹}, −∞<k<∞, be a sequence of iid random vectors in IR². We assume the existence of a sequence of positive constants a_n→∞, and a measure v on IR² which is finite on sets of the form {x : ‖x‖>r}, r>0 (where ‖⋅‖ denotes the Euclidean norm in IR²), such that nP{a_n ¹Z₀∈•}→^v v(•). (Here ‘→^v’ denotes vague convergence of measures on IR² with respect to the metric $d\left({\mathrm{x}}_1,{\mathrm{x}}_2\right)=\left|r_1^{-1}-r_2^{-1}\right|\vee \left|{\theta }_1-{\theta }_2\right|$, where for i=1,2,r, and θ_i denote the polar coordinates of x_j, and a∨b=max{a,b}.) The measure v is necessarily of the form v({x : ‖x‖>r,θ(x)∈A})=r^−aS(A) for r>0 and A⊂(0,2π), where S(•) is a probability measure on (0,2π) and α>0. Hence in particular [17].

$$v\left(cA\right)=c^{-\sigma }v\left(A\right)$$ (15)

for all c>0 and all sets A with v(A)<∞.

Define the bivariate moving average process ${\mathrm{X}}_n={\displaystyle {\sum }_{j-n}^{\infty }{C}_j{Z}_n}{}_j$, where ${\{C_j={\left[c_{jkl}\right]}_{kJ-1}^2\}}_{j\ge 0}$ is a sequence of real 2×2 matrices satisfying ${\displaystyle {\sum }_{j=0}^{\infty }{\left|C_{jkl}\right|}^{\mathrm{\delta }}<\infty ,k,l=1,2}$, for some δ∈(0,α). δ≤1. For x=(x₁,x₂)′∈IR², write A_xj= {z : C_jz∈((−∞,x₁)×(−∞,x₂))^c}, where A^c denotes the complement of a set A⊂IR². Then [16].

$$\begin{array}{c}\underset{x\to \infty }{\lim }P\left\{a_n^{-1}{\widehat{M}}_n\le \mathrm{x}\right\}=\exp \left\{-\widehat{\gamma }\left(\mathrm{x}\right)\right\},\text{and}\\ \underset{x\to \infty }{\lim }P\left\{a_n^{-1}{M}_n\le \mathrm{x}\right\}=\exp \left\{-\gamma \left(\mathrm{x}\right)\right\},x\in {\mathrm{IR}}^2,\end{array}$$

where $\widehat{\gamma }\left(\mathrm{x}\right)={\displaystyle {\sum }_{j=0}^{\infty }v\left(A_{xj}\right)}$ and $\gamma \left(\mathrm{x}\right)=v\left({\cup }_{j-0}^{\infty }{A}_{xj}\right)$. The extremal index is therefore $\theta \left(\mathrm{x}\right)=\gamma \left(\mathrm{x}\right)/\widehat{\gamma }\left(\mathrm{x}\right),\mathrm{x}\in {\mathrm{IR}}^2$. It follows from the definition of A_xj and Eq. (15) that this is in fact a function of x₁/x₂. Note that the extremal index defined above differs from that in Sec. 3 in that it is defined on IR² rather than [0,1]². However the two definitions arc equivalent as may be seen by means of a suitable transformation from IR² to [0,1]².

The actual calculation of θ(x) may be quite difficult in general, but possible to carry out under appropriate simplifying assumptions.

Case (i). If C_j=c_jC where $C={\left[C_{kt}\right]}_{kj-1}^2$, and the c_j’s are non-negative constants, then $A_{x_J}=C_J^{-1}B\left(\mathrm{x}\right)$ and ${\cup }_{j=0}^{\infty }{A}_{x_J}=cB\left(x\right)$, where B(x)={z : Cz∈((−∞,x₁)× (−∞,x₂))^c} and c=max{c_j : j≥0}. Therefore by Eq. (15), $v\left(A_{xj}\right)=c_j^{\alpha }v\left(B\left(\mathrm{x}\right)\right)$ and $v\left({\cup }_{j=0}^{\infty }{A}_{xj}\right)=c^{\alpha }v\left(B\left(\mathrm{x}\right)\right)$ so that $\theta \left(\mathrm{x}\right)\equiv c^{\sigma }/{\displaystyle {\sum }_{j=0}^{\infty }{c}_j^{\alpha }}$.

Case (ii). If the C_j’s are diagonal, i.e., C_j=diag[C_j1,C_j2] with c_ji≥0, i=1,2, then A_xj={z : c_jiz₁>x₁ or c_j2z₂>x₂} and ${\cup }_{j=0}^{\infty }{A}_{xj}={\left(\left(-\infty ,x_1/c_1\right)\times \left(-\infty x_2/c_2\right)\right)}^c$where c_i= max{c_ji : j≥0}, i=1,2. In particular, taking x₂=∞ and using Eq. (15) as in Case (i), we have $v\left(A_{xj}\right)=c_{ji}^{\alpha }v\left(\left\{z:z_1>x_1\right\}\right)$ and $v\left({\cup }_{j=0}^{\infty }{A}_{xj}\right)=c_1^{\alpha }v\left(\left\{z:z_1>x_1\right\}\right)$, so that the extremal index of ${\theta }_1=c_1^{\alpha }/{\displaystyle {\sum }_{j=0}^{\infty }{C}_{J-1}^{\alpha }}$.

Case (iii). Let D denote the support of v. If D⊂ {z : z₁=0 or z₂=0} (which is the case if the coordinates of Z₀ are independent), then we may write

$$v{\left(\left(-\infty ,x_1\right)\times \left(-\infty ,x_2\right)\right)}^c=a_1{x}_1^{-\alpha }+a_2{x_2}^{-\alpha },x_1,x_2\ge 0,$$ (16)

for suitable constants a₁≥0 and a₂≥0. Once again, assuming the c_jkl’s to be non-negative and writing c_ki=max{c_jki : j≥0} for k,l=1,2, we have (writing a∧b=min{a,b})

$$A_{1j}\cap D={\left(\left(-\infty ,x_1/c_{j,11}\wedge x_2/c_{j,21}\right)\times \left(-\infty ,x_1/c_{j,12}\wedge x_2/c_{j,22}\right)\right)}^c\cap D$$

and

$${\cup }_{j=0}^{\infty }{A}_{x,j}\cap D={\left(\left(-\infty ,x_1/c_{11}\wedge x_2/c_{21}\right)\times \left(-\infty ,x_1/c_{12}\wedge x_2/c_{22}\right)\right)}^c\cap D.$$

so that using Eq. (16)

$$\theta (\mathbf{x})=\frac{a_1{\left(x_1/c_{11}\wedge x_2/c_{21}\right)}^{-\alpha }+a_2{\left(x_1/c_{12}\wedge x_2/c_{22}\right)}^{-\alpha }}{a_1{\displaystyle {\sum }_{j=0}^{\infty }{\left(x_1/c_{j,11}\wedge x_2/c_{j,21}\right)}^{-\alpha }}+a_2{\displaystyle {\sum }_{j=0}^{\infty }{\left(x_1/c_{j,12}\wedge x_2/c_{j,22}\right)}^{-\alpha }}}$$

Thus putting x₂=∞, we have ${\theta }_1=\left(a_1{c}_{11}^{\alpha }+a_2{c}_{12}^{\alpha }\right)/\left(a_1{\displaystyle {\sum }_{j=0}^{\infty }{c}_{j,11}^{\alpha }}+a_2{\displaystyle {\sum }_{j=0}^{\infty }{c}_{j,12}^{\alpha }}\right)$ and similarly, ${\theta }_2=\left(a_1{c}_{21}^{\alpha }+a_2{c}_{22}^{\alpha }\right)/\left(a_1{\displaystyle {\sum }_{j=0}^{\infty }{c}_{j,21}^{\alpha }}+a_2{\displaystyle {\sum }_{j=0}^{\infty }{c}_{j,22}^{\alpha }}\right)$.

If also c_j,12=c_j,12=0 for each j, (that is if the C_j’s are diagonal), then

$$\theta (\mathbf{x})=\frac{a_1{x}_1^{-\alpha }{c}_{11}^{\alpha }+a_2{x}_2^{-\alpha }{c}_{22}^{\alpha }}{a_1{x}_1^{-\alpha }{\displaystyle {\sum }_{j=0}^{\alpha }{c}_{j,11}^{\alpha }}+a_2{x}_2^{-\alpha }{\displaystyle {\sum }_{j=0}^{\alpha }{c}_{j,22}^{\alpha }}},$$

and in particular, ${\theta }_1=c_{11}^{\alpha }/{\displaystyle {\sum }_{j=1}^{\infty }{c}_{j,11}^{\alpha }}$ and ${\theta }_2=c_{22}^{\alpha }/{\displaystyle {\sum }_{j=1}^{\infty }{c}_{j,22}^{\alpha }}$. Note that in this case the limiting distributions of M_n and ${\widehat{M}}_n$ are both independent, and hence (in accordance with Proposition 3.4) θ(x) is a convex combination of θ₁ and θ₂.

The non-negativeness of the C_j’s assumed abive is not crucial and may be related, although at the cost of more involved calculations.

Biography

About the author: S. Nandagopalan is an Assistant Professor of Statistics at Colorado State University.