Almost sure central limit theorem for self-normalized products of the some partial sums of ${\rho }^{-}$-mixing sequences

Abstract

Let ${\{X,X_n\}}_{n\in N}$ be a strictly stationary ${\rho }^{-}$-mixing sequence of positive random variables, under the suitable conditions, we get the almost sure central limit theorem for the products of the some partial sums ${(\frac{{\prod }_{i=1}^k{S}_{k,i}}{{(k-1)}^n{\mu }^n})}^{\frac{\mu }{\beta V_k}}$, where $\beta >0$ is a constant, and $\mathrm{E}(X)=\mu$, $S_{k,i}={\sum }_{j=1}^k{X}_j-X_i$, $1\le i\le k$, $V_k^2={\sum }_{i=1}^k{(X_i-\mu )}^2$.

Introduction and main result

In 1988, Brosamler [1] and Schatte [2] proposed the almost sure central limit theorem (ASCLT) for the sequence of i.i.d. random variables. On the basis of i.i.d., Khurelbaatar and Grzegorz [3] got the ASCLT for the products of the some partial sums of random variables. In 2008, Miao [4] gave a new form of ASCLT for products of some partial sums.

Theorem A

([4])

Let ${\{X,X_n\}}_{n\in N}$ be a sequence of i.i.d. positive square integrable random variables with $\mathrm{E}(X_1)=\mu$, $Var(X_1)={\sigma }^2>0$ and the coefficient of variation $\gamma =\frac{\sigma }{\mu }$. Denote the $S_{k,i}={\sum }_{j=1}^k{X}_j-X_i$, $1\le i\le k$. Then, for $\mathrm{\forall }x\in R$,

$$\underset{N\to \mathrm{\infty }}{lim}\frac{1}{logN}\sum _{n=1}^N\frac{1}{n}\mathrm{I}[{\left(\frac{{\prod }_{k=1}^n{S}_{n,k}}{{(n-1)}^n{\mu }^n}\right)}^{\frac{1}{\gamma \sqrt{n}}}\le x]=F(x)\mathit{\text{a.s.}},$$

where $F(\cdot )$ is the distribution function of the random variables $e^{\mathcal{N}}$, $\mathcal{N}$ is a standard normal random variable.

For random variables X, Y, define

$${\rho }^{-}(X,Y)=0\vee sup\frac{Cov(f(X),g(Y))}{{(Varf(X))}^{\frac{1}{2}}{(Varg(Y))}^{\frac{1}{2}}},$$

where the sup is taken over all $f,g\in \mathcal{C}$ such that $\mathrm{E}{(f(X))}^2<\mathrm{\infty }$ and $\mathrm{E}{(g(Y))}^2<\mathrm{\infty }$, and $\mathcal{C}$ is a class of functions which are coordinatewise increasing.

Definition

([5])

A sequence ${\{X,X_n\}}_{n\in N}$ is called ${\rho }^{-}$-mixing, if

$${\rho }^{-}(s)=sup\{{\rho }^{-}(S,T);S,T\subset N,dist(S,T)\ge s\}\to 0,s\to \mathrm{\infty },$$

where

$${\rho }^{-}(S,T)=0\vee sup\{\frac{Cov\{f(X_i,i\in S),g(X_j,j\in T)\}}{\sqrt{Var\{f(X_i,i\in S)\}Var\{g(X_j,j\in T)\}}},f,g\in \mathcal{C}\},$$

$\mathcal{C}$ is a class of functions which are coordinatewise increasing.

The precise definition of ${\rho }^{-}$-mixing random variables was introduced initially by Zhang and Wang [5] in 1999. Obviously, ${\rho }^{-}$-mixing random variables include NA and ${\rho }^{\ast }$-mixing random variables, which have a lot of applications, their limit properties have aroused wide interest recently, and a lot of results have been obtained by many authors. In 2005, Zhou [6] proved the almost central limit theorem of the ${\rho }^{-}$-mixing sequence. The almost sure central limit theorem for products of the partial sums of ${\rho }^{-}$-mixing sequences was given by Tan [7] in 2012. Because the denominator of the self-normalized partial sums contains random variables, this brings about difficulties to the study of the self-normalized form limit theorem of the ${\rho }^{-}$-mixing sequence. At present, there are very few results of this kind. In this paper, we extend Theorem A, and get the almost sure central limit theorem for self-normalized products of the some partial sums of ${\rho }^{-}$-mixing sequences.

Throughout this paper, $a_n\sim b_n$ means ${lim}_{n\to \mathrm{\infty }}\frac{a_n}{b_n}=1$, and C denotes a positive constant, which may take different values whenever it appears in different expressions, and $logx=ln(x\vee e)$. We assume ${\{X,X_n\}}_{n\in N}$ is a strictly stationary sequence of ${\rho }^{-}$-mixing random variables, and we denote $Y_i=X_i-\mu$.

For every $1\le i\le k\le n$, define

$$\begin{array}{c}{\overline{Y}}_{ni}=-\sqrt{n}\mathrm{I}(Y_i<-\sqrt{n})+Y_i\mathrm{I}(|Y_i|\le \sqrt{n})+\sqrt{n}\mathrm{I}(Y_i>\sqrt{n}),\hfill \\ T_{k,n}=\sum _{i=1}^k{\overline{Y}}_{ni},V_n^2=\sum _{i=1}^n{Y}_i^2,{\overline{V}}_n^2=\sum _{i=1}^n{\overline{Y}}_{ni}^2,\hfill \\ {\overline{V}}_{n,1}^2=\sum _{i=1}^n{\overline{Y}}_{ni}^2\mathrm{I}(Y_i\ge 0),{\overline{V}}_{n,2}^2=\sum _{i=1}^n{\overline{Y}}_{ni}^2\mathrm{I}(Y_i<0),\hfill \\ {\sigma }_n^2=Var(T_{n,n}),{\delta }_n^2=\mathrm{E}\left({\overline{Y}}_{n1}^2\right),{\delta }_{n,1}^2=\mathrm{E}{\overline{Y}}_{n1}^2\mathrm{I}(Y_1\ge 0),{\delta }_{n,2}^2=\mathrm{E}{\overline{Y}}_{n1}^2\mathrm{I}(Y_1<0),\hfill \end{array}$$

apparently, ${\delta }_n^2={\delta }_{n,1}^2+{\delta }_{n,2}^2$, $\mathrm{E}({\overline{V}}_n^2)=n{\delta }_n^2=n{\delta }_{n,1}^2+n{\delta }_{n,2}^2$.

Our main theorem is as follows.

Theorem 1

Let ${\{X,X_n\}}_{n\in N}$ be a strictly stationary ${\rho }^{-}$-mixing sequence of positive random variables with $\mathrm{E}X=\mu >0$, and for some $r>2$, we have $0<\mathrm{E}{|X|}^r<\mathrm{\infty }$. Denote $S_{k,i}={\sum }_{j=1}^k{X}_j-X_i$, $1\le i\le k$ and $Y=X-\mu$. Suppose that

(a₁)

$\mathrm{E}v(Y^2\mathrm{I}(Y\ge 0))>0$, $\mathrm{E}(Y^2\mathrm{I}(Y<0))>0$,

(a₂)

${\sigma }_1^2=\mathrm{E}{X}_1^2+2{\sum }_{k=2}^{\mathrm{\infty }}Cov(X_1,X_k)>0$, ${\sum }_{k=2}^{\mathrm{\infty }}|Cov(X_1,X_k)|<\mathrm{\infty }$,

(a₃)

${\sigma }_k^2\sim {\beta }^{2}k{\delta }_k^2$, for some $\beta >0$,

(a₄)

${\rho }^{-}(n)=O({log}^{-\delta }n)$, $\mathrm{\exists }\delta >1$.

Suppose $0\le \alpha <\frac{1}{2}$, and let

$$d_k=\frac{exp({log}^{\alpha }k)}{k},D_n=\sum _{k=1}^n{d}_k,$$ 1

then, for $\mathrm{\forall }x\in R$, we have

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k\mathrm{I}[{\left(\frac{{\prod }_{i=1}^k{S}_{k,i}}{{(k-1)}^k{\mu }^k}\right)}^{\frac{\mu }{\beta V_k}}\le x]=F(x)\mathit{\text{a.s.}},$$ 2

where $F(\cdot )$ is the distribution function of the random variables $e^{\mathcal{N}}$, $\mathcal{N}$ is a standard normal random variable.

Corollary 1

By [8], (2) remains valid if we replace the weight sequence $\{d_k,k\ge 1\}$ by any $\{d_k^{\ast },k\ge 1\}$ such that $0\le d_k^{\ast }\le d_k$, ${\sum }_{k=1}^{\mathrm{\infty }}{d}_k^{\ast }=\mathrm{\infty }$.

Corollary 2

If $\{X_n,n\ge 1\}$ is a sequence of strictly stationary independent positive random variables then one has (a₃) and $\beta =1$.

Some lemmas

We will need the following lemmas.

Lemma 2.1

([7])

Let ${\{X,X_n\}}_{n\in N}$ be a strictly stationary sequence of ${\rho }^{-}$-mixing random variables with $\mathrm{E}{X}_1=0$, $0<\mathrm{E}{X}_1^2<\mathrm{\infty }$, ${\sigma }_1^2=\mathrm{E}{X}_1^2+2{\sum }_{k=2}^{\mathrm{\infty }}Cov(X_1,X_k)>0$ and ${\sum }_{k=2}^{\mathrm{\infty }}|Cov(X_1,X_k)|<\mathrm{\infty }$, then, for $0<p<2$, we have

$$\frac{S_n}{n^{\frac{1}{p}}}\to 0,\mathit{\text{a.s.}},n\to \mathrm{\infty }.$$

Lemma 2.2

([9])

Let ${\{X,X_n\}}_{n\in N}$ be a sequence of ${\rho }^{-}$-mixing random variables, with

$$\mathrm{E}{X}_n=0,\mathrm{E}{|X_n|}^q<\mathrm{\infty },\mathrm{\forall }n\ge 1,q\ge 2,$$

then there is a positive constant $C=C(q,{\rho }^{-}(\cdot ))$ only depending on q and ${\rho }^{-}(\cdot )$ such that

$$\mathrm{E}\left(\underset{1\le j\le n}{max}{|S_j|}^q\right)\le C\{\sum _{i=1}^n\mathrm{E}{|X_i|}^q+{\left(\sum _{i=1}^n\mathrm{E}{X}_i^2\right)}^{\frac{q}{2}}\}.$$

Lemma 2.3

([10])

Suppose that $f_1(x)$ and $f_2(y)$ are real, bounded, absolutely continuous functions on R with $|f_1^{\prime }(x)|\le C_1$ and $|f_2^{\prime }(y)|\le C_2$, then, for any random variables X and Y,

$$|Cov(f_1(X),f_2(Y))|\le C_1{C}_2\{-Cov(X,Y)+8{\rho }^{-}(X,Y){\parallel X\parallel }_{2,1}{\parallel Y\parallel }_{2,1}\},$$

where ${\parallel X\parallel }_{2,1}={\int }_0^{\mathrm{\infty }}{(P(|X|>x))}^{\frac{1}{2}}dx$.

Lemma 2.4

Let ${\{\xi ,{\xi }_n\}}_{n\in N}$ be a sequence of uniformly bounded random variables. If $\mathrm{\exists }\delta >1$, ${\rho }^{-}(n)=O({log}^{-\delta }n)$, there exist constants $C>0$ and $\epsilon >0$, such that

$$|\mathrm{E}{\xi }_k{\xi }_l|\le C({\rho }^{-}(k)+{\left(\frac{k}{l}\right)}^{\epsilon }),1\le 2k<l,$$ 3

then

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k{\xi }_k=0,\mathit{\text{a.s.}}$$

Proof

See the proof of Theorem 1 in [7]. □

Lemma 2.5

If the assumptions of Theorem 1 hold, then

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k\mathrm{I}[\frac{T_{k,k}-\mathrm{E}(T_{k,k})}{\beta {\delta }_k\sqrt{k}}\le x]=\mathrm{\Phi }(x)\mathit{\text{a.s.}},\mathrm{\forall }x\in R,$$ 4

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k[f\left(\frac{{\overline{V}}_{k,l}^2}{k{\delta }_{k,l}^2}\right)-\mathrm{E}f\left(\frac{{\overline{V}}_{k,l}^2}{k{\delta }_{k,l}^2}\right)]=0\mathit{\text{a.s.}},l=1,2,$$ 5

where $d_k$ and $D_k$ is defined as (1) and f is real, bounded, absolutely continuous function on R.

Proof

Firstly, we prove (4), by the property of ${\rho }^{-}$-mixing sequence, we know that ${\{{\overline{Y}}_{ni}\}}_{n\ge 1,i\le n}$ is a ${\rho }^{-}$-mixing sequence; using Lemma 2.1 in [7], the condition (a₂), (a₃), and $\beta >0$, ${\delta }_k^2\to \mathrm{E}{Y}^2>0$, it follows that

$$\frac{T_{k,k}-\mathrm{E}(T_{k,k})}{\beta {\delta }_k\sqrt{k}}\stackrel{\mathrm{d}}{\to }\mathcal{N},k\to \mathrm{\infty },$$

hence, for any $g(x)$ which is a bounded function with bounded continuous derivative, we have

$$\mathrm{E}g\left(\frac{T_{k,k}-E(T_{k,k})}{\beta {\delta }_k\sqrt{k}}\right)\to \mathrm{E}g(\mathcal{N}),k\to \mathrm{\infty },$$

by the Toeplitz lemma, we get

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k\mathrm{E}\left[g\left(\frac{T_{k,k}-\mathrm{E}(T_{k,k})}{\beta {\delta }_k\sqrt{k}}\right)\right]=\mathrm{E}(g(\mathcal{N})).$$

On the other hand, from Theorem 7.1 of [11] and Sect. 2 of [12], we know that (4) is equivalent to

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_{k}g\left(\frac{T_{k,k}-\mathrm{E}(T_{k,k})}{\beta {\delta }_k\sqrt{k}}\right)=\mathrm{E}(g(\mathcal{N}))\text{a.s.},$$

hence, to prove (4), it suffices to prove

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k[g\left(\frac{T_{k,k}-\mathrm{E}(T_{k,k})}{\beta {\delta }_k\sqrt{k}}\right)-\mathrm{E}\left(g\frac{T_{k,k}-\mathrm{E}(T_{k,k})}{\beta {\delta }_k\sqrt{k}}\right)]=0\text{a.s.},$$ 6

noting that

$${\xi }_k=g\left(\frac{T_{k,k}-\mathrm{E}(T_{k,k})}{\beta {\delta }_k\sqrt{k}}\right)-\mathrm{E}\left(g\left(\frac{T_{k,k}-\mathrm{E}(T_{k,k})}{\beta {\delta }_k\sqrt{k}}\right)\right),$$

for every $1\le 2k<l$, we have

$$\begin{array}{rcl}|\mathrm{E}{\xi }_k{\xi }_l|& =& |Cov(g\left(\frac{T_{k,k}-\mathrm{E}{T}_{k,k}}{\beta {\delta }_k\sqrt{k}}\right),g\left(\frac{T_{l,l}-\mathrm{E}{T}_{l,l}}{\beta {\delta }_l\sqrt{l}}\right))|\\ & \le & |Cov(g\left(\frac{T_{k,k}-\mathrm{E}{T}_{k,k}}{\beta {\delta }_k\sqrt{k}}\right),g\left(\frac{T_{l,l}-\mathrm{E}{T}_{l,l}}{\beta {\delta }_l\sqrt{l}}\right)-g\left(\frac{T_{l,l}-\mathrm{E}{T}_{l,l}-(T_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta {\delta }_l\sqrt{l}}\right))|\\ & & +|Cov(g\left(\frac{T_{k,k}-\mathrm{E}{T}_{k,k}}{\beta {\delta }_k\sqrt{k}}\right),g\left(\frac{T_{l,l}-\mathrm{E}{T}_{l,l}-(T_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta {\delta }_l\sqrt{l}}\right))|\\ & =& I_1+I_2.\end{array}$$ 7

First we estimate $I_1$; we know that g is a bounded Lipschitz function, i.e., there exists a constant C such that

$$|g(x)-g(y)|\le C|x-y|$$

for any $x,y\in R$, since ${\{{\overline{Y}}_{ni}\}}_{n\ge 1,i\le n}$ also is a ${\rho }^{-}$-mixing sequence; we use the condition ${\delta }_l^2\to \mathrm{E}(Y^2)<\mathrm{\infty }$, $l\to \mathrm{\infty }$, and Lemma 2.2, to get

$$\begin{array}{rcl}{I}_1& \le & C\frac{\mathrm{E}|T_{2k,l}-\mathrm{E}{T}_{2k,l}|}{\sqrt{l}}\le C\frac{\sqrt{\mathrm{E}{(T_{2k,l}-\mathrm{E}{T}_{2k,l})}^2}}{\sqrt{l}}\\ & \le & \frac{C}{\sqrt{l}}\sqrt{\sum _{i=1}^{2k}\mathrm{E}{\overline{Y}}_{l,i}^2}\le \frac{C}{\sqrt{l}}\sqrt{\sum _{i=1}^{2k}\mathrm{E}{Y}^2}\le C{\left(\frac{k}{l}\right)}^{\frac{1}{2}}.\end{array}$$ 8

Next we estimate $I_2$; by Lemma 2.2, we have

$$\begin{array}{rl}Var\left(\frac{T_{k,k}-\mathrm{E}{T}_{k,k}}{\beta {\delta }_k\sqrt{k}}\right)& \le \frac{C}{k}Var(T_{k,k}-\mathrm{E}{T}_{k,k})\\ & \le \frac{C}{k}\sum _{i=1}^k\mathrm{E}{({\overline{Y}}_{ki}-\mathrm{E}{\overline{Y}}_{ki})}^2\le \frac{C}{k}\sum _{i=1}^k\mathrm{E}{({\overline{Y}}_{ki})}^2\le \frac{C}{k}\cdot k\le C\end{array}$$

and

$$\begin{array}{rl}Var\left(\frac{T_{l,l}-\mathrm{E}{T}_{l,l}-(T_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta {\delta }_l\sqrt{l}}\right)& \le \frac{C}{l}Var(T_{l,l}-\mathrm{E}{T}_{l,l}-(T_{2k,l}-\mathrm{E}{T}_{2k,l}))\\ & \le \frac{C}{l}\sum _{i=2k+1}^l\mathrm{E}{({\overline{Y}}_{li}-\mathrm{E}{\overline{Y}}_{li})}^2\le \frac{C}{l}\left(\sum _{i=1}^l\mathrm{E}{\overline{Y}}_{li}^2\right)\\ & \le \frac{C}{l}\cdot l\le C.\end{array}$$

By the definition of a ${\rho }^{-}$-mixing sequence, $\mathrm{E}{Y}^2<\mathrm{\infty }$, and Lemma 2.3, we have

$$\begin{array}{rcl}{I}_2& \le & (-Cov(\frac{T_{k,k}-\mathrm{E}{T}_{k,k}}{\beta {\delta }_k\sqrt{k}},\frac{T_{l,l}-\mathrm{E}{T}_{l,l}-(T_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta {\delta }_l\sqrt{l}})\\ & & +8{\rho }^{-}(\frac{T_{k,k}-\mathrm{E}{T}_{k,k}}{\beta {\delta }_k\sqrt{k}},\frac{T_{l,l}-\mathrm{E}{T}_{l,l}-(T_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta {\delta }_l\sqrt{l}})\\ & & \cdot {\parallel \frac{T_{k,k}-\mathrm{E}{T}_{k,k}}{\beta {\delta }_k\sqrt{k}}\parallel }_{2,1}\cdot {\parallel \frac{T_{l,l}-\mathrm{E}{T}_{l,l}-(T_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta {\delta }_l\sqrt{l}}\parallel }_{2,1})\\ & \le & C{\rho }^{-}(k){(Var\left(\frac{T_{k,k}-\mathrm{E}{T}_{k,k}}{\beta {\delta }_k\sqrt{k}}\right))}^{\frac{1}{2}}\cdot {(Var\left(\frac{T_{l,l}-\mathrm{E}{T}_{l,l}-(T_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta {\delta }_l\sqrt{l}}\right))}^{\frac{1}{2}}\\ & & +8{\rho }^{-}(k)\cdot {\parallel \frac{T_{k,k}-\mathrm{E}{T}_{k,k}}{\beta {\delta }_k\sqrt{k}}\parallel }_{2,1}\cdot {\parallel \frac{T_{l,l}-\mathrm{E}{T}_{l,l}-(T_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta {\delta }_l\sqrt{l}}\parallel }_{2,1}.\end{array}$$

By ${\parallel X\parallel }_{2,1}\le r/(r-2){\parallel X\parallel }_r$, $r>2$ (see p. 254 of [10] or p. 251 of [13]), Minkowski inequality, Lemma 2.2, and the Hölder inequality, we get

$$\begin{array}{rcl}{\parallel \frac{T_{k,k}-\mathrm{E}{T}_{k,k}}{\beta {\delta }_k\sqrt{k}}\parallel }_{2,1}& \le & \frac{r}{r-2}{\parallel \frac{T_{k,k}-\mathrm{E}{T}_{k,k}}{\beta {\delta }_k\sqrt{k}}\parallel }_r\\ & =& \frac{r}{r-2}\frac{1}{\beta {\delta }_k\sqrt{k}}{(\mathrm{E}|T_{k,k}-\mathrm{E}{T}_{k,k}{|}^r)}^{\frac{1}{r}}\\ & \le & \frac{C}{\sqrt{k}}{(\sum _{i=1}^k\mathrm{E}{|{\overline{Y}}_{ki}|}^r+{\left(\sum _{i=1}^k\mathrm{E}{\overline{Y}}_{ki}^2\right)}^{r/2})}^{1/r}\\ & \le & \frac{C}{\sqrt{k}}{(k+k^{r/2})}^{1/r}\le C,\end{array}$$

similarly

$${\parallel \frac{T_{l,l}-\mathrm{E}{T}_{l,l}-(T_{2k,l}-\mathrm{E}{T}_{2k,l})}{\beta {\delta }_l\sqrt{l}}\parallel }_{2,1}\le C.$$

Hence

$$I_2\le C{\rho }^{-}(k).$$ 9

Combining with (7)–(9), (3) holds, and by (a₄), Lemma 2.4, (6) holds, then (4) is true.

Secondly, we prove (5); for $\mathrm{\forall }k\ge 1$, ${\eta }_k=f({\overline{V}}_{k,1}^2/(k{\delta }_{k,1}^2))-\mathrm{E}(f({\overline{V}}_{k,1}^2/(k{\delta }_{k,1}^2)))$, we have

$$\begin{array}{rcl}|\mathrm{E}{\eta }_k{\eta }_l|& =& |Cov(f\left(\frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2}\right),f\left(\frac{{\overline{V}}_{l,1}^2}{l{\delta }_{l,1}^2}\right))|\\ & \le & |Cov(f\left(\frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2}\right),f\left(\frac{{\overline{V}}_{l,1}^2}{l{\delta }_{l,1}^2}\right)-f\left(\frac{{\sum }_{i=2k+1}^l{\overline{Y}}_{l,i}^{2}I(Y_i\ge 0)}{l{\delta }_{l,1}^2}\right))|\\ & & +|Cov(f\left(\frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2}\right),f\left(\frac{{\sum }_{i=2k+1}^l{\overline{Y}}_{l,i}^{2}I(Y_i\ge 0)}{l{\delta }_{l,1}^2}\right))|\\ & =& J_1+J_2,\end{array}$$ 10

by the property of f, we know

$$J_1\le C(\mathrm{E}(\sum _{i=1}^{2k}{\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0))/l)\le C\left(\frac{k}{l}\right).$$ 11

Now we estimate $J_2$,

$$\begin{array}{rcl}Var\left(\frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2}\right)& =& Var\left(\frac{{\sum }_{i=1}^k{\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0)}{k{\delta }_{k,1}^2}\right)\\ & \le & \frac{C}{k^2}\mathrm{E}{(\sum _{i=1}^k{\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0))}^2\\ & =& \frac{C}{k^2}\mathrm{E}{(\sum _{i=1}^k{\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0)-\mathrm{E}(\sum _{i=1}^k{\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0))+\mathrm{E}(\sum _{i=1}^k{\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0)))}^2\\ & \le & \frac{C}{k^2}\mathrm{E}{\left(\sum _{i=1}^k({\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0)-\mathrm{E}({\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0)))\right)}^2\\ & & +\frac{C}{k^2}{\left(\sum _{i=1}^k\mathrm{E}({\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0))\right)}^2\\ & \le & \frac{C}{k^2}\sum _{i=1}^k\mathrm{E}{\overline{Y}}_{ki}^4\mathrm{I}(Y_i\ge 0)+\frac{C}{k^2}{\left(k\mathrm{E}({\overline{Y}}_{k1}^2\mathrm{I}(Y_1\ge 0))\right)}^2\\ & \le & \frac{C}{k^2}\sum _{i=1}^k\mathrm{E}k{(Y_i)}^2\le C,\end{array}$$

and similarly $Var({\sum }_{i=2k+1}^l{\overline{Y}}_{li}^2\mathrm{I}(Y_i\ge 0)/(l{\delta }_{l,1}^2))\le C$. On the other hand, we have

$$\begin{array}{rcl}{\parallel \frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2}\parallel }_{2,1}& \le & \frac{r}{r-2}\cdot \frac{C}{k}{\left(\mathrm{E}\right|{\overline{V}}_{k,1}^2{|}^r)}^{1/r}\\ & \le & \frac{C}{k}{\left(\mathrm{E}\right|\sum _{i=1}^k({\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0)-\mathrm{E}({\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0))){|}^r+\left|\sum _{i=1}^k\mathrm{E}({\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0)){|}^r\right)}^{1/r}\\ & \le & \frac{C}{k}\left(\sum _{i=1}^k\mathrm{E}\right|({\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0)-\mathrm{E}({\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0))){|}^r\\ & & +{\left(\sum _{i=1}^k\mathrm{E}{({\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0)-\mathrm{E}({\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0)))}^2\right)}^{r/2}{)}^{1/r}\\ & & +\frac{C}{k}\left|\sum _{i=1}^k\mathrm{E}({\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0))\right|\\ & \le & \frac{C}{k}{\left(\sum _{i=1}^k\mathrm{E}\right|{\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0){|}^r+{\left(\sum _{i=1}^k\mathrm{E}\right|{\overline{Y}}_{ki}^2\mathrm{I}(Y_i\ge 0){|}^2)}^{r/2})}^{1/r}\\ & & +\frac{C}{k}\left|k\mathrm{E}({\overline{Y}}_{k1}^2\mathrm{I}(Y_1\ge 0))\right|\\ & \le & \frac{C}{k}{(\sum _{i=1}^k\mathrm{E}|\sqrt{k}{Y}_i{|}^r+{(\sum _{i=1}^k\mathrm{E}|\sqrt{k}{Y}_i{|}^2)}^{r/2})}^{1/r}+C_1\\ & \le & \frac{C}{k}{(k^{1+r/2}+k^r)}^{1/r}+C_1\le C,\end{array}$$

similarly

$${\parallel \sum _{i=2k+1}^l{\overline{Y}}_{li}^2\mathrm{I}(Y_i\ge 0)/\left(l{\delta }_{l,1}^2\right)\parallel }_{2,1}\le C.$$

Thus, by Lemma 2.3, we have

$$\begin{array}{rcl}{J}_2& \le & C\{-Cov(\frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2},\frac{{\sum }_{i=2k+1}^l{\overline{Y}}_{li}^2\mathrm{I}(Y_i\ge 0)}{l{\delta }_{l,1}^2})\\ & & +8{\rho }^{-}(\frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2},\frac{{\sum }_{i=2k+1}^l{\overline{Y}}_{li}^2\mathrm{I}(Y_i\ge 0)}{l{\delta }_{l,1}^2})\cdot {\parallel \frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2}\parallel }_{2,1}\cdot {\parallel \frac{{\sum }_{i=2k+1}^l{\overline{Y}}_{li}^2\mathrm{I}(Y_i\ge 0)}{l{\delta }_{l,1}^2}\parallel }_{2,1}\}\\ & \le & C\{{\rho }^{-}(k){(Var\left(\frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2}\right))}^{1/2}\cdot Var{\left(\frac{{\sum }_{i=2k+1}^l{\overline{Y}}_{li}^2\mathrm{I}(Y_i\ge 0)}{l{\delta }_{l,1}^2}\right)}^{1/2}\\ & & +{\rho }^{-}(k)\cdot {\parallel \frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2}\parallel }_{2,1}\cdot {\parallel \frac{{\sum }_{i=2k+1}^l{\overline{Y}}_{li}^2\mathrm{I}(Y_i\ge 0)}{l{\delta }_{l,1}^2}\parallel }_{2,1}\}\\ & \le & C{\rho }^{-}(k),\end{array}$$ 12

hence, combining with (11) and (12), (3) holds, and by Lemma 2.4, (5) holds. □

Proof of Theorem 1

Let $C_{k,i}=\frac{S_{k,i}}{(k-1)\mu }$, hence, (2) is equivalent to

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k\mathrm{I}(\frac{\mu }{\beta V_k}\sum _{i=1}^{k}log{C}_{k,i}\le x)=\mathrm{\Phi }(x)\text{a.s.}$$ 13

So we only need to prove (13), for a fixed k, $1\le k\le n$ and $\mathrm{\forall }\epsilon >0$; we have

$$\underset{k\to \mathrm{\infty }}{lim}P\left\{\bigcup _{m=k}^{\mathrm{\infty }}\left(\right|\frac{X_i}{m}|\ge \epsilon )\right\}=\underset{k\to \mathrm{\infty }}{lim}P\left\{\right|\frac{X_i}{k}|\ge \epsilon \}=\underset{k\to \mathrm{\infty }}{lim}P\{|X_1|\ge \epsilon k\}=0,$$

therefore, by Theorem 1.5.2 in [14], we have

$$\frac{X_i}{k}\to 0\text{a.s. }k\to \mathrm{\infty },$$

on the unanimous establishment of i.

By Lemma 2.1, for some $\frac{4}{3}<p<2$, and enough large k, we have

$$\begin{array}{rcl}\underset{1\le i\le k}{sup}|C_{k,i}-1|& \le & \left|\frac{{\sum }_{j=1}^k(X_j-\mu )}{(k-1)\mu }\right|+\underset{1\le i\le k}{sup}\left|\frac{X_i}{(k-1)\mu }\right|+\frac{1}{k-1}\\ & \le & |\frac{S_k-k\mu }{k^{\frac{1}{p}}}\cdot \frac{k^{\frac{1}{p}}}{(k-1)\mu }|\le C{k}^{\frac{1}{p}-1},\end{array}$$

by $log(1+x)=x+O(x^2)$, $x\to 0$, we get

$$\begin{array}{rl}& |\frac{\mu }{\beta {\delta }_k\sqrt{(1\pm \epsilon )k}}\sum _{i=1}^{k}ln{C}_{k,i}-\frac{\mu }{\beta {\delta }_k\sqrt{(1\pm \epsilon )k}}\sum _{i=1}^k(C_{k,i}-1)|\\ & \le \frac{C\mu }{\beta {\delta }_k\sqrt{(1\pm \epsilon )k}}\sum _{i=1}^k{(C_{k,i}-1)}^2\\ & \le \frac{C}{\sqrt{k}}{k}^{\frac{2}{p}-1}\to 0\text{a.s.},k\to \mathrm{\infty },\end{array}$$

and then, for $\delta >0$ and every ω, there exists $k_0=k_0(\omega ,\delta ,x)$; when $k>k_0$, we have

$$\begin{array}{rl}& \mathrm{I}\{\frac{\mu }{\beta {\delta }_k\sqrt{(1\pm \epsilon )k}}\sum _{i=1}^k(C_{k,i}-1)\le x-\delta \}\\ & \le \mathrm{I}\{\frac{\mu }{\beta {\delta }_k\sqrt{(1\pm \epsilon )k}}\sum _{i=1}^{k}log{C}_{k,i}\le x\}\\ & \le \mathrm{I}\{\frac{\mu }{\beta {\delta }_k\sqrt{(1\pm \epsilon )k}}\sum _{i=1}^k(C_{k,i}-1)\le x+\delta \},\end{array}$$ 14

under the condition $|X_i-\mu |\le \sqrt{k}$, $1\le i\le k$, we have

$$\mu \sum _{i=1}^k(C_{k,i}-1)=\sum _{i=1}^k\frac{S_{k,i}-(k-1)\mu }{k-1}=\sum _{i=1}^k{Y}_i=\sum _{i=1}^k{\overline{Y}}_{ki}=T_{k,i},$$ 15

furthermore, by (14) and (15), for any given $0<\epsilon <1$, $\delta >0$, when $k>k_0$, we obtain

$$\begin{array}{c}\mathrm{I}(\frac{\mu }{\beta V_k}\sum _{i=1}^{k}log{C}_{k,i}\le x)\hfill \\ \le \mathrm{I}(\frac{T_{k,i}}{{\delta }_k\beta \sqrt{k(1+\epsilon )}}\le x+\delta )+\mathrm{I}({\overline{V}}_k^2>(1+\epsilon )k{\delta }_k^2)\hfill \\ +\mathrm{I}\left(\bigcup _{i=1}^k(|X_i-\mu |>\sqrt{k})\right),x\ge 0,\hfill \\ \mathrm{I}(\frac{\mu }{\beta V_k}\sum _{i=1}^{k}log{C}_{k,i}\le x)\hfill \\ \le \mathrm{I}(\frac{T_{k,i}}{{\delta }_k\beta \sqrt{k(1-\epsilon )}}\le x+\delta )+\mathrm{I}({\overline{V}}_k^2<(1-\epsilon )k{\delta }_k^2)\hfill \\ +\mathrm{I}\left(\bigcup _{i=1}^k(|X_i-\mu |>\sqrt{k})\right),x<0,\hfill \\ \mathrm{I}(\frac{\mu }{\beta V_k}\sum _{i=1}^{k}log{C}_{k,i}\le x)\hfill \\ \ge \mathrm{I}(\frac{T_{k,i}}{{\delta }_k\beta \sqrt{k(1-\epsilon )}}\le x-\delta )-\mathrm{I}({\overline{V}}_k^2<(1-\epsilon )k{\delta }_k^2)\hfill \\ -\mathrm{I}\left(\bigcup _{i=1}^k(|X_i-\mu |>\sqrt{k})\right),x\ge 0,\hfill \\ \mathrm{I}(\frac{\mu }{\beta V_k}\sum _{i=1}^{k}log{C}_{k,i}\le x)\hfill \\ \ge \mathrm{I}(\frac{T_{k,i}}{{\delta }_k\beta \sqrt{k(1+\epsilon )}}\le x-\delta )-\mathrm{I}({\overline{V}}_k^2>(1+\epsilon )k{\delta }_k^2)\hfill \\ -\mathrm{I}\left(\bigcup _{i=1}^k(|X_i-\mu |>\sqrt{k})\right),x<0.\hfill \end{array}$$

Therefore, to prove (13), for any $0<\epsilon <1$, ${\delta }_1>0$, it suffices to prove

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k\mathrm{I}(\frac{T_{k,i}}{\beta {\delta }_k\sqrt{k}}\le \sqrt{1\pm \epsilon }x\pm {\delta }_1)=\mathrm{\Phi }(\sqrt{1\pm \epsilon }x\pm {\delta }_1)\text{a.s.},$$ 16

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k\mathrm{I}\left(\bigcup _{i=1}^k(|X_i-\mu |>\sqrt{k})\right)=0\text{a.s.},$$ 17

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k\mathrm{I}({\overline{V}}_k^2>(1+\epsilon )k{\delta }_k^2)=0\text{a.s.},$$ 18

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k\mathrm{I}({\overline{V}}_k^2<(1-\epsilon )k{\delta }_k^2)=0\text{a.s.}$$ 19

Firstly, we prove (16), by $\mathrm{E}(Y^2)<\mathrm{\infty }$, we know ${lim}_{x\to \mathrm{\infty }}{x}^{2}P(|Y|>x)=0$, and by $\mathrm{E}(Y)=0$, it follows that

$$\begin{array}{rcl}|\mathrm{E}(T_{k,i})|& =& \left|\mathrm{E}\left(\sum _{i=1}^k{\overline{Y}}_{ki}\right)\right|=|k\mathrm{E}{\overline{Y}}_{k1}|\\ & \le & k\left|\mathrm{E}\left(Y\mathrm{I}(|Y|>\sqrt{k})\right)\right|+k^{\frac{3}{2}}\mathrm{E}\left(\mathrm{I}(|Y|>\sqrt{k})\right)\\ & \le & \sqrt{k}\mathrm{E}\left(Y^2\mathrm{I}(|Y|>\sqrt{k})\right)+k^{\frac{3}{2}}P(|Y|>\sqrt{k})=o(\sqrt{k}),\end{array}$$

so, combining with ${\delta }_k^2\to \mathrm{E}(Y^2)<\mathrm{\infty }$, for any $\alpha >0$, when $k\to \mathrm{\infty }$, we have

$$\begin{array}{rl}& \mathrm{I}(\frac{T_{k,i}-E{T}_{k,i}}{\beta {\delta }_k\sqrt{k}}\le \sqrt{1\pm \epsilon }x\pm {\delta }_1-\alpha )\\ & \le \mathrm{I}(\frac{T_{k,i}}{\beta {\delta }_k\sqrt{k}}\le \sqrt{1\pm \epsilon }x\pm {\delta }_1)\\ & \le \mathrm{I}(\frac{T_{k,i}-E{T}_{k,i}}{\beta {\delta }_k\sqrt{k}}\le \sqrt{1\pm \epsilon }x\pm {\delta }_1+\alpha ),\end{array}$$

thus, by (4), we get

$$\begin{array}{rl}& \underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k\mathrm{I}(\frac{T_{k,i}}{\beta {\delta }_k\sqrt{k}}\le \sqrt{1\pm \epsilon }x\pm {\delta }_1)\\ & \ge \underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k\mathrm{I}(\frac{T_{k,i}-E{T}_{k,i}}{\beta {\delta }_k\sqrt{k}}\le \sqrt{1\pm \epsilon }x\pm {\delta }_1-\alpha )\\ & \to \mathrm{\Phi }(\sqrt{1\pm \epsilon }x\pm {\delta }_1-\alpha ),\end{array}$$ 20

$$\begin{array}{rl}& \underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k\mathrm{I}(\frac{T_{k,i}}{\beta {\delta }_k\sqrt{k}}\le \sqrt{1\pm \epsilon }x\pm {\delta }_1)\\ & \le \underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k\mathrm{I}(\frac{T_{k,i}-E{T}_{k,i}}{\beta {\delta }_k\sqrt{k}}\le \sqrt{1\pm \epsilon }x\pm {\delta }_1+\alpha )\\ & \to \mathrm{\Phi }(\sqrt{1\pm \epsilon }x\pm {\delta }_1+\alpha )\text{a.s.},\end{array}$$ 21

letting $\alpha \to 0$ in (20) and (21), (16) holds.

Now, we prove (17); by $\mathrm{E}(Y^2)<\mathrm{\infty }$, we know ${lim}_{x\to \mathrm{\infty }}{x}^{2}P(|Y|>x)=0$, such that

$$\mathrm{E}\mathrm{I}\left(\bigcup _{i=1}^k(|Y_i|>\sqrt{k})\right)\le \sum _{i=1}^{k}P(|Y_i|>\sqrt{k})\le kP(|Y|>\sqrt{k})\to 0,k\to \mathrm{\infty },$$

by the Toeplitz lemma, we get

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k\mathrm{E}\mathrm{I}\left(\bigcup _{i=1}^k(|Y_i|>\sqrt{k})\right)\to 0\text{a.s.},$$ 22

hence, to prove (17), it suffices to prove

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k(\mathrm{I}\left(\bigcup _{i=1}^k(|Y_i|>\sqrt{k})\right)-\mathrm{E}\left[\mathrm{I}\left(\bigcup _{i=1}^k(|Y_i|>\sqrt{k})\right)\right])\to 0\text{a.s.},$$ 23

writing

$${\mathcal{Z}}_k=\mathrm{I}\left(\bigcup _{i=1}^k(|Y_i|>\sqrt{k})\right)-\mathrm{E}\left[\mathrm{I}\left(\bigcup _{i=1}^k(|Y_i|>\sqrt{k})\right)\right],$$

for every $0\le 2k<l$, so by the definition of ${\rho }^{-}$-mixing sequence, we have

$$\begin{array}{rcl}\mathrm{E}|{\mathcal{Z}}_k{\mathcal{Z}}_l|& =& |Cov(\mathrm{I}\left(\bigcup _{i=1}^k(|Y_i|>\sqrt{k})\right),\mathrm{I}\left(\bigcup _{i=1}^l(|Y_i|>\sqrt{l})\right))|\\ & \le & |Cov(\mathrm{I}\left(\bigcup _{i=1}^k(|Y_i|>\sqrt{k})\right),\mathrm{I}\left(\bigcup _{i=1}^l(|Y_i|>\sqrt{l})\right)-\mathrm{I}\left(\bigcup _{i=2k+1}^l(|Y_i|>\sqrt{l})\right))|\\ & & +|Cov(\mathrm{I}\left(\bigcup _{i=1}^k(|Y_i|>\sqrt{k})\right),\mathrm{I}\left(\bigcup _{i=2k+1}^l(|Y_i|>\sqrt{l})\right))|\\ & \le & \mathrm{E}|\mathrm{I}\left(\bigcup _{i=1}^l(|Y_i|>\sqrt{l})\right)-\mathrm{I}\left(\bigcup _{i=2k+1}^l(|Y_i|>\sqrt{l})\right)|\\ & & +{\rho }^{-}(k)\sqrt{Var\left(\mathrm{I}\left(\bigcup _{i=1}^k(|Y_i|>\sqrt{k})\right)\right)Var\left(\mathrm{I}\left(\bigcup _{i=2k+1}^l(|Y_i|>\sqrt{l})\right)\right)}\\ & \le & \mathrm{E}\left[\mathrm{I}\left(\bigcup _{i=1}^{2k}(|Y_i|>\sqrt{l})\right)\right]+C{\rho }^{-}(k)\\ & \le & \sum _{i=1}^{k}P(|Y_i|>\sqrt{l})+C{\rho }^{-}(k)\\ & \le & kP(|Y|>\sqrt{l})+C{\rho }^{-}(k)\\ & \le & C(\frac{k}{l}+{\rho }^{-}(k)),\end{array}$$

so by Lemma 2.4, (23) holds. And combining with (22), we know that (17) holds.

Next, we prove (18); by $\mathrm{E}({\overline{V}}_k^2)=k{\delta }_k^2$, ${\overline{V}}_k^2={\overline{V}}_{k,1}^2+{\overline{V}}_{k,2}^2$, $\mathrm{E}({\overline{V}}_{k,l}^2)=k{\delta }_{k,l}^2$, and ${\delta }_{k,1}^2\le {\delta }_k^2$, $l=1,2$, we have

$$\begin{array}{rcl}\mathrm{I}({\overline{V}}_k^2>(1+\epsilon )k{\delta }_k^2)& =& \mathrm{I}({\overline{V}}_k^2-\mathrm{E}\left({\overline{V}}_k^2\right)>\epsilon k{\delta }_k^2)\\ & \le & \mathrm{I}({\overline{V}}_{k,1}^2-\mathrm{E}\left({\overline{V}}_{k,1}^2\right)>\epsilon k{\delta }_k^2/2)+\mathrm{I}({\overline{V}}_{k,2}^2-\mathrm{E}\left({\overline{V}}_{k,2}^2\right)>\epsilon k{\delta }_k^2/2)\\ & \le & \mathrm{I}({\overline{V}}_{k,1}^2>(1+\frac{\epsilon }{2})k{\delta }_{k,1}^2)+\mathrm{I}({\overline{V}}_{k,2}^2>(1+\frac{\epsilon }{2})k{\delta }_{k,2}^2),\end{array}$$

therefore, by the arbitrariness of $\epsilon >0$, to prove (18), it suffices to prove

$$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{D_n}\sum _{k=1}^n{d}_k\mathrm{I}({\overline{V}}_{k,l}^2>(1+\frac{\epsilon }{2})k{\delta }_{k,l}^2)=0\text{a.s. }l=1,2,$$ 24

when $l=1$, for given $\epsilon >0$, let f be a bounded function with bounded continuous derivative such that

$$\mathrm{I}(x>1+\epsilon )\le f(x)\le \mathrm{I}(x>1+\frac{\epsilon }{2}),$$ 25

under the condition

$$\mathrm{E}\left({\overline{V}}_{k,1}^2\right)=k{\delta }_{k,1}^2,\mathrm{E}\left(Y^2\right)<\mathrm{\infty },\mathrm{E}(Y^2\mathrm{I}(Y\ge 0))>0,$$

by the Markov inequality, and Lemma 2.2, we get

$$\begin{array}{rl}& P({\overline{V}}_{k,1}^2>(1+\frac{\epsilon }{2})k{\delta }_{k,1}^2)\\ & =P({\overline{V}}_{k,1}^2-\mathrm{E}\left({\overline{V}}_{k,1}^2\right)>\frac{\epsilon }{2}k{\delta }_{k,1}^2)\\ & \le C\frac{\mathrm{E}{({\overline{V}}_{k,1}^2-\mathrm{E}({\overline{V}}_{k,1}^2))}^2}{k^2}\le C\frac{{\sum }_{i=1}^k\mathrm{E}{({\overline{Y}}_{ki}^2\mathrm{I}({\overline{Y}}_{ki}\ge 0))}^2}{k^2}\\ & \le C\frac{\mathrm{E}{\overline{Y}}_{k1}^4\mathrm{I}({\overline{Y}}_{k1}\ge 0)}{k}\le C\frac{\mathrm{E}{Y}^4\mathrm{I}(0\le Y\le \sqrt{k})+k^{2}P(Y>\sqrt{k})}{k},\end{array}$$ 26

because $\mathrm{E}(Y^2)<\mathrm{\infty }$ implies ${lim}_{x\to \mathrm{\infty }}{x}^{2}P(|Y|>x)=0$, we have

$$\begin{array}{rcl}\mathrm{E}{Y}^4\mathrm{I}(0\le Y\le \sqrt{k})& =& {\int }_0^{\mathrm{\infty }}P(|Y|\mathrm{I}(0\le Y\le \sqrt{k})\ge t)4t^{3}dt\\ & \le & C{\int }_0^{\sqrt{k}}P(|Y|\ge t)t^{3}dt\\ & =& {\int }_0^{\sqrt{k}}o(1)tdt=o(1)k,\end{array}$$

thus, combining with (26),

$$P({\overline{V}}_{k,1}^2>(1+\frac{\epsilon }{2})k{\delta }_{k,1}^2)\to 0,k\to \mathrm{\infty }.$$

Therefore, from (5), (25) and the Toeplitz lemma

$$\begin{array}{rcl}0& \le & \frac{1}{D_n}\sum _{k=1}^n{d}_k\mathrm{I}({\overline{V}}_{k,1}^2>(1+\frac{\epsilon }{2})k{\delta }_{k,1}^2)\\ & \le & \frac{1}{D_n}\sum _{k=1}^n{d}_{k}f\left(\frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2}\right)\\ & =& \frac{1}{D_n}\sum _{k=1}^n{d}_k\mathrm{E}\left(f\left(\frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2}\right)\right)+\frac{1}{D_n}\sum _{k=1}^n{d}_k(f\left(\frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2}\right)-\mathrm{E}\left(f\left(\frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2}\right)\right))\\ & \le & \frac{1}{D_n}\sum _{k=1}^n{d}_k\mathrm{E}\left(\mathrm{I}({\overline{V}}_{k,1}^2>(1+\frac{\epsilon }{2})k{\delta }_{k,1}^2)\right)+\frac{1}{D_n}\sum _{k=1}^n{d}_k(f\left(\frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2}\right)-\mathrm{E}\left(f\left(\frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2}\right)\right))\\ & =& \frac{1}{D_n}\sum _{k=1}^n{d}_{k}P({\overline{V}}_{k,1}^2>(1+\frac{\epsilon }{2})k{\delta }_{k,1}^2)+\frac{1}{D_n}\sum _{k=1}^n{d}_k(f\left(\frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2}\right)-\mathrm{E}\left(f\left(\frac{{\overline{V}}_{k,1}^2}{k{\delta }_{k,1}^2}\right)\right))\\ & \to & 0\text{a.s.},k\to \mathrm{\infty },\end{array}$$

hence, (24) holds for $l=1$. Similarly, we can prove (24) for $l=2$, so (18) is true. By similar methods used to prove (18), we can prove (19), this completes the proof of Theorem 1.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Funding

This work was supported by the National Natural Science Foundation of China (11171003), the Foundation of Jilin Educational Committee of China (2015-155) and the Innovation Talent Training Program of Science and Technology of Jilin Province of China (20180519011JH).

Competing interests

The authors declare that there is no conflict of interest regarding the publication of this paper. We confirm that the received funding mentioned in the “Acknowledgment” section did not lead to any conflict of interests regarding the publication of this manuscript. We declare that we do not have any commercial or associated interest that represents a conflict of interest in connection with the work submitted.