An almost sure central limit theorem of products of partial sums for ρ^--mixing sequences

Abstract

Let {X _n , n ≥ 1} be a strictly stationary ρ^--mixing sequence of positive random variables with EX₁ = μ > 0 and Var(X₁) = σ² < ∞. Denote $S_n=\sum _{i=1}^n{X}_i$ and $\gamma =\frac{\sigma }{\mu }$ the coefficient of variation. Under suitable conditions, by the central limit theorem of weighted sums and the moment inequality we show that

$$\forall x=\underset{n\to \infty }{\text{lim}}\frac{1}{\text{log}n}\sum _{k=1}^n\frac{1}{k}I\left\{{\left(\prod _{i=1}^k\frac{S_i}{i\mu }\right)}^{\frac{1}{\gamma \sigma k}}\right\}=F\left(x\right)a.s.,$$

where ${\sigma }_k^2=Var\left(S_{k,k}\right),S_{k,k}=\sum _{i=1}^k{b}_{i,k}{Y}_i,b_{i,k}=\sum _{j=i}^k\frac{1}{j},i\le k$ with $b_{i,k}=0,i>k,Y_i=\frac{X_{i-\mu }}{\sigma },F\left(x\right)$ is the distribution function of the random variable $e^{\sqrt{2}\mathcal{N}}$, and $\mathcal{N}$ is a standard normal random variable.

MR(2000) Subject Classification: 60F15.

1 Introduction and main results

For a random variable X, define ∥X∥ _p = (E|X|^p)^1/p. For two nonempty disjoint sets S,T ⊂ N, we define dist(S,T) to be min{|j - k|; j ∈ S, k ∈ T}. Let σ(S) be the σ-field generated by {X _k , k ∈ S}, and define σ(T) similarly. Let $C$ be a class of functions which are coordinatewise increasing. For any real number x, x⁺, and x^- denote its positive and negative part, respectively, (except for some special definitions, for examples, ρ^-(s), ρ^-(S,T), etc.). For random variables X, Y, define

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

where the sup is taken over all $f,g\in C$ such that E(f(X))² < ∞ and E(g(Y))² < ∞.

A sequence {X _n , n ≥ 1} is called negatively associated (NA) if for every pair of disjoint subsets S, T of N,

$$Cov\left\{f\left(X_i,i\in S\right),g\left(X_j,j\in T\right)\right\}\le 0,$$

whenever $f,g\in C$.

A sequence {X _n , n ≥ 1} is called ρ*-mixing if

$$\rho *\left(s\right)=\text{sup}\left\{\rho \left(S,T\right);S,T\subset N,\text{dist}\left(S,T\right)\ge s\right\}\to 0ass\to \infty ,$$

where

$$\rho \left(S,T\right)=\text{sup}\left\{\left|E\left(f-Ef\right)\left(g-Eg\right)/\left({∥f-Ef∥}_2\cdot {∥g-Eg∥}_2\right)\right|;f\in L_2\left(\sigma \left(S\right)\right),g\in L_2\left(\sigma \left(T\right)\right)\right\}.$$

A sequence {X _n , n ≥ 1} is called ρ^--mixing, if

$${\rho }^{\text{\_}}\left(s\right)=\text{sup}\left\{{\rho }^{\text{\_}}\left(S,T\right);S,T\subset N,\text{dist}\left(S,T\right)\ge s\right\}\to 0ass\to \infty .$$

where,

$${\rho }^{\text{\_}}\left(S,T\right)=0\vee \text{sup}\left\{\frac{Cov\left\{f\left(X_i,i\in S\right),g\left(X_i,j\in T\right)\right\}}{\sqrt{Var\left\{f\left(X_i,i\in S\right)\right\}Var\left\{g\left(X_i,j\in T\right)\right\}}};f,g\in C\right\}.$$

The concept of ρ^--mixing random variables was proposed in 1999 (see [1]). Obviously, ρ^--mixing random variables include NA and ρ*-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, such as the weak convergence theorems, the central limit theorems of random fields, Rosenthal-type moment inequality, see [1–4]. Zhou [5] studied the almost sure central limit theorem of ρ^--mixing sequences by the conditions provided by Shao: on the conditions of central limit theorem, and if ${\epsilon }_0>0,Var\left(\sum _{i=1}^n\frac{1}{i}f\left(\frac{S_i}{{\sigma }_i}\right)\right)=O\left({\text{log}}^{2-{\epsilon }_0}n\right)$, where f is Lipschitz function. In this article, we study the almost sure central limit theorem of products of partial sums for ρ^--mixing sequence by the central limit theorem of weighted sums and moment inequality.

Here and in the sequel, let $b_{k,n}=\sum _{i=k}^n\frac{1}{i},k\le n$ with b_k,n= 0, k > n. Suppose {X _n , n ≥ 1} be a strictly stationary ρ^--mixing sequence of positive random variables with EX₁ = μ > 0 and Var(X₁) = σ² < ∞. Let ${\tilde{S}}_n=\sum _{k=1}^n{Y}_k$ and $S_{n,n}=\sum _{k=1}^n{b}_{k,n}{Y}_k$, where $Y_k=\frac{X_k-\mu }{\sigma },k\ge 1$. Let ${\sigma }_n^2=Var\left(S_{n,n}\right)$, and C denotes a positive constant, which may take different values whenever it appears in different expressions. The following are our main results.

Theorem 1.1 Let {X _n , n ≥ 1} be a defined as above with 0 < E|X₁|^r< ∞ for a certain r > 2, denote $S_n=\sum _{i=1}^n{X}_i$ and $\gamma =\frac{\sigma }{\mu }$ the coefficient of variation. Assume that

(a₁) ${\sigma }_1^2=E{X}_1^2+2\sum _{n=2}^{\infty }Cov\left(X_1,X_n\right)>0$,

(a₂) $\sum _{n=2}^{\infty }\left|Cov\left(X_1,X_n\right)\right|<\infty$,

(a₃) ρ^-(n) = O(log^-δn), ∃ δ > 1,

(a₄) $\underset{n\in N}{\text{inf}}\frac{{\sigma }_n^2}{n}>0$.

Then

$$\forall x\underset{n\to \infty }{\text{lim}}\frac{1}{\text{log}n}\sum _{k=1}^n\frac{1}{k}I\left\{\frac{S_{k,k}}{{\sigma }_k}\le x\right\}=\Phi \left(x\right)a.s.$$

(1.1)

Here and in the sequel, I{·} denotes indicator function and Φ(·) is the distribution function of standard normal random variable $\mathcal{N}$.

Theorem 1.2 Under the conditions of Theorem 1.1, then

$$\forall x=\underset{n\to \infty }{\text{lim}}\frac{1}{\text{log}n}\sum _{k=1}^n\frac{1}{k}I\left\{{\left(\prod _{i=1}^k\frac{S_i}{i\mu }\right)}^{\frac{1}{\gamma \sigma k}}\le x\right\}=F\left(x\right)a.s.$$

(1.2)

Here and in the sequel, F(·) is the distribution function of the random variable $e^{\sqrt{2}\mathcal{N}}$.

2 Some lemmas

To prove our main results, we need the following lemmas.

Lemma 2.1 [3] Let {X _n , n ≥ 1} be a weakly stationary ρ^--mixing sequence with $E{X}_n=0,0<E{X}_1^2<\infty$, and

• (i) ${\sigma }_1^2=E{X}_1^2+2\sum _{n=2}^{\infty }Cov\left(X_1,X_n\right)>0$,

• (ii) $\sum _{n=2}^{\infty }\left|Cov\left(X_1,X_n\right)\right|<\infty$,

then

$$\frac{E{S}_n^2}{n}\to {{\sigma }_1}^2,\frac{S_n}{{\sigma }_1\sqrt{n}}\stackrel{d}{\to }\mathcal{N}\left(0,1\right)asn\to \infty .$$

Lemma 2.2 [4] For a positive real number q ≥ 2, if {X _n , n ≥ 1} is a sequence of ρ^--mixing random variables with EX _i = 0, E|X _i |^q< ∞ for every i ≥ 1, then for all n ≥ 1, there is a positive constant C = C(q, ρ^-(·)) such that

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

Lemma 2.3 [6] $\sum _{i=1}^n{b}_{i,n}^2=2n-b_{1,n}$.

Lemma 2.4 [[3], Theorem 3.2] Let {X _ni , 1 ≤ i ≤ n, n ≥ 1} be an array of centered random variables with $E{X}_{ni}^2<\infty$ for each i = 1,2,...,n. Assume that they are ρ^--mixing. Let {a _ni , 1 ≤ i ≤ n, n ≥ 1} be an array of real numbers with a _ni = ±1 for i = 1, 2,..., n. Denote ${\sigma }_n^2=Var\left(\sum _{i=1}^n{a}_{ni}{X}_{ni}\right)$ and suppose that

$$\underset{n}{\text{sup}}\frac{1}{{\sigma }_n^2}\sum _{i=1}^{n}E{X}_{ni}^2<\infty ,$$

and

$$\underset{n\to \infty }{\lim \sup }\frac{1}{{\sigma }_n^2}{\displaystyle \sum _{\underset{1\le i,j\le n}{i,j:\left|i-j\right|\ge A}}Cov{\left(X_{ni},X_{nj}\right)}^{-}\to 0asA\to \infty ,}$$

and the following Lindeberg condition is satisfied:

$$\frac{1}{{\sigma }_n^2}\sum _{i=1}^{n}E{X}_{ni}^{2}I\left\{\left|X_{ni}\right|\ge \epsilon {\sigma }_n\right\}\to 0asn\to \infty$$

for every ε > 0. Then

$$\frac{1}{{\sigma }_n}\sum _{i=1}^n{a}_{ni}{X}_{ni}\stackrel{d}{\to }\mathcal{N}\left(0,1\right)asn\to \infty .$$

Lemma 2.5 Let {X _n , n ≥ 1} be a strictly stationary sequence of ρ^--mixing random variables with EX _n = 0 and $\sum _{n=2}^{\infty }\left|Cov\left(X_1,X_n\right)\right|<\infty ,\left\{a_{ni},1\le i\le n,n\ge 1\right\}$ be an array of real numbers such that $\underset{n}{\text{sup}}\sum _{i=1}^n{a}_{ni}^2<\infty$ and $\underset{1\le i\le n}{\text{max}}\left|a_{ni}\right|\to 0$ as n → ∞. If $Var\left(\sum _{i=1}^n{a}_{ni}{X}_i\right)=1$ and $\left\{X_n^2\right\}$ is an uniformly integrable family, then

$$\sum _{i=1}^n{a}_{ni}{X}_i\stackrel{d}{\to }\mathcal{N}\left(0,1\right)asn\to \infty .$$

Proof Notice that

$$\sum _{i=1}^n{a}_{ni}{X}_i=\sum _{i=1}^n\frac{a_{ni}}{\left|a_{ni}\right|}\left|a_{ni}\right|X_i=:\sum _{i=1}^n{b}_{ni}{Y}_{ni},$$

where $b_{ni}=\frac{a_{ni}}{\left|a_{ni}\right|}$ and Y _ni = |a _ni |X _i . Then {Y _ni , 1 ≤ i ≤ n, n ≥ 1} is an array of ρ^--mixing centered random variables with $E{Y}_{ni}^2=a_{ni}^{2}E{X}_i^2<\infty$ and b _ni = ±1 for i = 1, 2,..., n and ${\sigma }_n^2=Var\left(\sum _{i=1}^n{b}_{ni}{Y}_{ni}\right)=1$. Note that $\left\{X_n^2\right\}$ is an uniformly integrable family, we have

$$\underset{n}{\text{sup}}\frac{1}{{\sigma }_n^2}\sum _{i=1}^{n}E{Y}_{ni}^2=\underset{n}{\text{sup}}\sum _{i=1}^n{a}_{ni}^{2}E{X}_i^2\le \underset{n}{\text{sup}}\sum _{i=1}^n{a}_{ni}^2\cdot \underset{i}{\text{sup}}E{X}_i^2<\infty ,$$

and

$$\begin{array}{l}\underset{n\to \infty }{\text{lim sup}}\frac{1}{{\sigma }_n^2}{\displaystyle \sum _{\underset{1\le i,j\le n}{i,j:\left|i-j\right|\ge A}}Cov{\left(Y_{ni},Y_{nj}\right)}^{-}}\\ =\underset{n\to \infty }{\lim \sup }{\displaystyle \sum _{\underset{1\le i,j\le n}{i,j:\left|i-j\right|\ge A}}Cov{\left(\left|a_{ni}\right|X_i,\left|a_{nj}\right|X_j\right)}^{-}}\\ \le \underset{n\to \infty }{\lim \sup }{\displaystyle \sum _{\underset{1\le i,j\le n}{i,j:\left|i-j\right|\ge A}}\left|a_{ni}\right|\cdot \left|a_{nj}\right|\cdot \left|Cov\left(X_i,X_j\right)\right|}\\ \le C\left(\underset{n\to \infty }{\lim \sup }\left({\displaystyle \sum _{\underset{1\le i,j\le n}{i,j:\left|i-j\right|\ge A}}{\left|a_{ni}\right|}^2\cdot \left|Cov\left(X_i,X_j\right)\right|}+{\displaystyle \sum _{\underset{1\le i,j\le n}{i,j:\left|i-j\right|\ge A}}{\left|a_{ni}\right|}^2\cdot \left|Cov\left(X_i,X_j\right)\right|}\right)\right)\\ \le C\underset{n}{\sup }{\displaystyle \sum _{i=1}^n{\left|a_{ni}\right|}^2}\cdot {\displaystyle \sum _{i>A}\left|Cov(X_1,X_i)\right|}\\ \to 0asA\to \infty ,\end{array}$$

and ∀ ε > 0, we get

$$\begin{array}{l}\frac{1}{{\sigma }_n^2}\sum _{i=1}^{n}E{Y}_{ni}^{2}I\left\{\left|Y_{ni}\right|\ge \epsilon {\sigma }_n\right\}\\ =\sum _{i=1}^n{a}_{ni}^{2}E{X}_i^{2}I\left\{\left|a_{ni}\right|\cdot \left|X_i\right|\ge \epsilon \right\}\\ \le \underset{n}{\text{sup}}\sum _{i=1}^n{a}_{ni}^2\cdot E{X}_1^{2}I\left\{\left|a_{ni}\right|\cdot \left|X_1\right|\ge \epsilon \right\}\\ \le \underset{n}{\text{sup}}\sum _{i=1}^n{a}_{ni}^2\cdot E{X}_1^{2}I\left\{\underset{1\le i\le n}{\text{max}}\left|a_{ni}\right|\cdot \left|X_1\right|\ge \epsilon \right\}\to 0asn\to \infty ,\end{array}$$

thus the conclusion is proved by Lemma 2.4.

Lemma 2.6 [2] Suppose that f₁(x) and f₂(y) are real, bounded, absolutely continuous functions on R with $\left|{f^{\prime }}_1\left(x\right)\right|\le C_1$ and $\left|{f^{\prime }}_2\left(y\right)\right|\le C_2$. Then for any random variables X and Y,

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

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

Lemma 2.7 Let {X _n , n ≥ 1} be a strictly stationary sequence of ρ^--mixing random variables with $E{X}_1=0,0<E{X}_1^2<\infty$ and

$$\begin{array}{ll}\hfill 0<{\sigma }_1^2& =E{X}_1^2+2\sum _{n=2}^{\infty }Cov\left(X_1,X_n\right)<\infty ,\\ \sum _{n=2}^{\infty }\left|Cov\left(X_1,X_n\right)\right|<\infty ,\end{array}$$

then for 0 < p < 2, we have

$$n^{-\frac{1}{p}}{S}_n\to 0a.s.asn\to \infty .$$

Proof By Lemma 2.1, we have

$$\underset{n\to \infty }{\text{lim}}\frac{E{S}_n^2}{n}={\sigma }_1^2.$$

(2.1)

Let n _k = k^α, where $\alpha >\text{max}\left\{1,\frac{p}{2-p}\right\}$. By (2.1), we get

$$\sum _{k=1}^{\infty }P\left\{\left|S_{nk}\right|\ge \epsilon n_k^{\frac{1}{p}}\right\}\le \sum _{k=1}^{\infty }\frac{E{S}_{n_k}^2}{{\epsilon }^2{n}_k^{\frac{2}{p}}}\le \sum _{k=1}^{\infty }\frac{C}{{\epsilon }^2{k}^{\alpha \left(\frac{2}{p}-1\right)}}<\infty .$$

From Borel-Cantelli lemma, it follows that

$$n_k^{-\frac{1}{p}}{S}_{n_k}\to 0a.s.ask\to \infty .$$

(2.2)

And by Lemma 2.2, it follows that

$$\begin{array}{l}\sum _{k=1}^{\infty }P\left\{\underset{n_k\le n<n_{k+1}}{\text{max}}\frac{\left|S_n-S_{n_k}\right|}{n^{\frac{1}{p}}}\ge \epsilon \right\}\le \sum _{k=1}^{\infty }\frac{E\underset{n_k\le n<n_{k+1}}{\text{max}}{\left|S_n-S_{n_k}\right|}^2}{{\epsilon }^2{n}_k^{\frac{2}{p}}}\\ =\sum _{k=1}^{\infty }\frac{E\underset{n_k\le n<n_{k+1}}{\text{max}}{\left|\sum _{i=n_{k+1}}^n{X}_i\right|}^2}{{\epsilon }^2{n}_k^{\frac{2}{p}}}\le C\sum _{k=1}^{\infty }\frac{\left(n_{k+1}-n_k\right)}{{\epsilon }^2{n}_k^{\frac{2}{p}}}\le C\sum _{k=1}^{\infty }\frac{1}{k^{\alpha \left(\frac{2}{p}-1\right)}}<\infty .\end{array}$$

By Borel-Cantelli lemma, we conclude that

$$\underset{n_k\le n<n_{k+1}}{\text{max}}\frac{\left|S_n-S_{n_k}\right|}{n^{\frac{1}{p}}}\to 0a.s.asn\to \infty .$$

(2.3)

For every n, there exist n _k and n_k+1such that n _k ≤ n < n_k+1, by (2.2) and (2.3), we have

$$\begin{array}{ll}\hfill \frac{\left|S_n\right|}{n^{\frac{1}{p}}}& =\frac{\left|S_n-S_{n_k}+S_{n_k}\right|}{n^{\frac{1}{p}}}\\ \le \frac{\left|S_{n_k}\right|}{n_k^{\frac{1}{p}}}+\underset{n_k\le n<n_{k+1}}{\text{max}}\frac{\left|S_n-S_{n_k}\right|}{n^{\frac{1}{p}}}\to 0a.s.asn\to \infty .\end{array}$$

The proof is now completed.

3 Proof of the theorems

Proof of Theorem 1.1 By the property of ρ^--mixing sequence, it is easy to see that {Y _n } is a strictly stationary ρ^--mixing sequence with EY₁ = 0 and $E{Y}_1^2=1$. We first prove

$$\frac{S_{n,n}}{{\sigma }_n}\stackrel{d}{\to }\mathcal{N}\left(0,1\right)asn\to \infty .$$

(3.1)

Let $a_{ni}=\frac{b_{i,n}}{{\sigma }_n},1\le i\le n,n\ge 1$. Obviously,

$$Var\left(\sum _{i=1}^n{a}_{ni}{Y}_i\right)=1.$$

From condition (a₄) in Theorem 1.1 and Lemma 2.3, we have

$$\underset{n}{\text{sup}}\sum _{i=1}^n{a}_{ni}^2=\underset{n}{\text{sup}}\sum _{i=1}^n\frac{b_{i,n}^2}{{\sigma }_n^2}=\underset{n}{\text{sup}}\frac{2n-b_{1,n}}{{\sigma }_n^2}\le C\underset{n}{\text{sup}}\frac{2n-b_{1,n}}{n}<\infty ,$$

and

$$\underset{1\le i\le n}{\text{max}}\left|a_{ni}\right|=\underset{1\le i\le n}{\text{max}}\frac{b_{i,n}}{{\sigma }_n}\le \frac{b_{1,n}}{{\sigma }_n}\le \frac{C\text{log}n}{\sqrt{n}}\to 0asn\to \infty .$$

By stationarity of {Y _n , n ≥ 1} and E |X₁|² < ∞, we know that $\left\{Y_n^2\right\}$ is uniformly integrable, and from condition (a₂) in Theorem 1.1, we get $\sum _{n=2}^{\infty }\left|Cov\left(Y_1,Y_n\right)\right|<\infty$, so applying Lemma 2.5, we have

$$\sum _{i=1}^n{a}_{ni}{Y}_i\stackrel{d}{\to }\mathcal{N}\left(0,1\right).$$

Notice that

$$\sum _{i=1}^n{a}_{ni}{Y}_i=\sum _{i=1}^n\frac{b_{i,n}{Y}_i}{{\sigma }_n}=\frac{S_{n,n}}{{\sigma }_n},$$

so (3.1) is valid. Let f(x) be a bounded Lipschitz function and have a Radon-Nikodyn derivative h(x) bounded by Γ. From (3.1), we have

$$Ef\left(\frac{S_{n,n}}{{\sigma }_n}\right)\to Ef\left(\mathcal{N}\left(0,1\right)\right)asn\to \infty ,$$

thus

$$\frac{1}{\text{log}n}\sum _{k=1}^n\frac{1}{k}Ef\left(\frac{S_{k,k}}{{\sigma }_k}\right)-Ef\left(\mathcal{N}\left(0,1\right)\right)\to 0asn\to \infty .$$

(3.2)

On the other hand, note that (1.1) is equivalent to

$$\underset{n\to \infty }{\text{lim}}\frac{1}{\text{log}n}\sum _{k=1}^n\frac{1}{k}f\left(\frac{S_{k,k}}{{\sigma }_k}\right)={\int }_{-\infty }^{\infty }f\left(x\right)d\Phi \left(x\right)=Ef\left(\mathcal{N}\left(0,1\right)\right)a.s.$$

(3.3)

from Section 2 of Peligrad and Shao [7] and Theorem 7.1 on P₄₂ from Billingsley [8]. Hence, to prove (3.3), it suffices to show that

$$T_n=\frac{1}{\text{log}n}\sum _{k=1}^n\frac{1}{k}\left[f\left(\frac{S_{k,k}}{{\sigma }_k}\right)-Ef\left(\frac{S_{k,k}}{{\sigma }_k}\right)\right]\to 0a.s.n\to \infty$$

(3.4)

by (3.2). Let ${\xi }_k=f\left(\frac{S_{k,k}}{{\sigma }_k}\right)-Ef\left(\frac{S_{k,k}}{{\sigma }_k}\right)$, 1 ≤ k ≤ n m we have

$$\begin{array}{ll}\hfill E{T}_n^2& =\frac{1}{{\text{log}}^{2}n}E{\left(\sum _{k=1}^n\frac{{\xi }_k}{k}\right)}^2\\ \le \frac{1}{{\text{log}}^{2}n}\sum _{1\le k\le l\le n,2k\ge l}\frac{\left|E{\xi }_k{\xi }_l\right|}{kl}+\frac{1}{{\text{log}}^{2}n}\sum _{1\le k\le l\le n,2k\ge l}\frac{\left|E{\xi }_k{\xi }_l\right|}{kl}\\ :=I_1+I_2.\end{array}$$

(3.5)

By the fact that f is bounded, we have

$$I_1\le \frac{C}{{\text{log}}^{2}n}\sum _{k=1}^n\sum _{l=k}^{2k}\frac{1}{kl}=\frac{C}{{\text{log}}^{2}n}\sum _{k=1}^n\frac{1}{k}\sum _{l=k}^{2k}\frac{1}{l}\le C\left({\text{log}}^{-1}n\right).$$

(3.6)

Now we estimate I₂, if l > 2k, we have

$$\begin{array}{ll}\hfill S_{l,l}-S_{2k,2k}& =\left(b_{1,l}{Y}_1+b_{2,l}{Y}_2+\cdots +b_{l,l}{Y}_l\right)-\left(b_{1,2k}{Y}_1+b_{2,2k}{Y}_2+\cdots +b_{2k,2k}{Y}_{2k}\right)\\ =\left(b_{2k+1,l}{Y}_{2k+1}+\cdots +b_{l,l}{Y}_l\right)+b_{2k+1,l}{\tilde{S}}_{2k},\end{array}$$

and

$$\begin{array}{ll}\hfill \left|E{\xi }_k{\xi }_l\right|& =\left|Cov\left(f\left(\frac{S_{k,k}}{{\sigma }_k}\right),f\left(\frac{S_{l,l}}{{\sigma }_l}\right)\right)\right|\\ \le \left|Cov\left(f\left(\frac{S_{k,k}}{{\sigma }_k}\right),f\left(\frac{S_{l,l}}{{\sigma }_l}\right)-f\left(\frac{S_{l,l}-S_{2k,2k}-b_{2k+1,l}{\tilde{S}}_{2k}}{{\sigma }_l}\right)\right)\right|\\ +\left|Cov\left(f\left(\frac{S_{k,k}}{{\sigma }_k}\right),f\left(\frac{S_{l,l}-S_{2k,2k}-b_{2k+1,l}{\tilde{S}}_{2k}}{{\sigma }_l}\right)\right)\right|.\end{array}$$

By Lemma 2.3 and condition (a₂) in Theorem 1.1, we have

$$\begin{array}{ll}\hfill Var\left(S_{k,k}\right)& =\sum _{i=1}^k{b}_{i,k}^{2}E{Y}_i^2+2\sum _{j=1}^{k-1}\sum _{i=j+1}^k{b}_{i,k}{b}_{j,k}Cov\left(Y_i,Y_j\right)\\ \le \sum _{i=1}^k{b}_{i,k}^2+2\sum _{j=1}^k{b}_{j,k}^2\sum _{i=j+1}^k\left|Cov\left(Y_i,Y_j\right)\right|\\ \le Ck,\end{array}$$

(3.7)

and

$$\begin{array}{l}Var\left(S_{l,l}-S_{2k,2k}-b_{2k+1,l}{\tilde{S}}_{2k}\right)\\ =\sum _{i=2k+1}^l{b}_{i,l}^{2}E{Y}_i^2+2\sum _{j=2k+1}^{l-1}\sum _{i=j+1}^l{b}_{i,l}{b}_{j,l}Cov\left(Y_i,Y_j\right)\\ \le \sum _{i=2k+1}^l{b}_{i,l}^2+2\sum _{j=1}^l{b}_{i,l}^2\sum _{i=j+1}^l\left|Cov\left(Y_i,Y_j\right)\right|\\ \le Cl.\end{array}$$

By Lemma 2.6, the definition of ρ^--mixing sequence and condition (a₄), we have

$$\begin{array}{l}\left|Cov\left(f\left(\frac{S_{k,k}}{{\sigma }_k}\right),f\left(\frac{S_{l,l}-S_{2k,2k}-b_{2k+1,l}{\tilde{S}}_{2k}}{{\sigma }_l}\right)\right)\right|\\ \le C\left\{-Cov\left(\frac{S_{k,k}}{{\sigma }_k},\frac{S_{l,l}-S_{2k,2k}-b_{2k+1,l}{\tilde{S}}_{2k}}{{\sigma }_l}\right)\right.\\ \left.+8{\rho }^{-}\left(\frac{S_{k,k}}{{\sigma }_k},\frac{S_{l,l}-S_{2k,2k}-b_{2k+1,l}{\tilde{S}}_{2k}}{{\sigma }_l}\right)\cdot {∥\frac{S_{k,k}}{{\sigma }_k}∥}_{2,1}\cdot {∥\frac{S_{l,l}-S_{2k,2k}-b_{2k+1,l}{\tilde{S}}_{2k}}{{\sigma }_l}∥}_{2,1}\right\}\\ \le C{\rho }^{-}\left(k\right){\left(Var\left(\frac{S_{k,k}}{{\sigma }_k}\right)\right)}^{\frac{1}{2}}{\left(Var\left(\frac{S_{l,l}-S_{2k,2k}-b_{2k+1,l}{\tilde{S}}_{2k}}{{\sigma }_l}\right)\right)}^{\frac{1}{2}}\\ +C{\rho }^{-}\left(k\right){∥\frac{S_{k,k}}{{\sigma }_k}∥}_{2,1}\cdot {∥\frac{S_{l,l}-S_{2k,2k}-b_{2k+1,l}{\tilde{S}}_{2k}}{{\sigma }_l}∥}_{2,1}\\ \le C{\rho }^{-}\left(k\right)+C{\rho }^{-}\left(k\right){∥\frac{S_{k,k}}{{\sigma }_k}∥}_{2,1}\cdot {∥\frac{S_{l,l}-S_{2k,2k}-b_{2k+1,l}{\tilde{S}}_{2k}}{{\sigma }_l}∥}_{2,1}.\end{array}$$

By the inequality ${∥X∥}_{2,1}\le \frac{r}{r-2}{∥X∥}_r\left(r>2\right)$ (cf. Zhang [[2], p. 254] or Ledoux and Talagrand [[9], p. 251]), we get

$${∥\frac{S_{k,k}}{{\sigma }_k}∥}_{2,1}\le \frac{r}{r-2}{∥\frac{S_{k,k}}{{\sigma }_k}∥}_r=\frac{r}{r-2}\frac{1}{{\sigma }_k}{\left(E{\left|S_{k,k}\right|}^r\right)}^{\frac{1}{r}},$$

and

$$\begin{array}{ll}\hfill E{\left|S_{k,k}\right|}^r& =E|\sum _{j=1}^k{b}_{j,k}{Y}_j{|}^r\\ \le C\left\{\sum _{j=1}^k{b}_{j,k}^{r}E{\left|X_j\right|}^r+{\left(\sum _{j=1}^k{b}_{j,k}^{2}E{X}_j^2\right)}^{\frac{r}{2}}\right\}\\ \le C\left\{k\left({\text{log}}^{r}k\right)+k^{\frac{r}{2}}\right\},\end{array}$$

thus

$${∥\frac{S_{k,k}}{{\sigma }_k}∥}_{2,1}\le C\left(\frac{r}{r-2}\cdot \frac{\text{log}k}{k^{\frac{1}{2}-\frac{1}{r}}}+\frac{r}{r-2}\right)<C,$$

similarly,

$${∥\frac{S_{l,l}-S_{2k,2k}-b_{2k+1,l}{\tilde{S}}_{2k}}{{\sigma }_l}∥}_{2,1}<C,$$

hence

$$\left|Cov\left(f\left(\frac{S_{k,k}}{{\sigma }_k}\right),f\left(\frac{S_{l,l}-S_{2k,2k}-b_{2k+1,l}{\tilde{S}}_{2k}}{{\sigma }_l}\right)\right)\right|\le C{\rho }^{-}\left(k\right).$$

Similarly to (3.7), we have

$$\begin{array}{ll}\hfill Var\left(S_{2k,2k}\right)& =\sum _{i=1}^{2k}{b}_{i,2k}^{2}E{Y}_i^2+2\sum _{j=1}^{2k-1}\sum _{i=j+1}^{2k}{b}_{i,2k}{b}_{j,2k}Cov\left(Y_i,Y_j\right)\\ \le \sum _{i=1}^{2k}{b}_{i,2k}^2+2\sum _{j=1}^{2k-1}{b}_{i,2k}^2\sum _{i=j+1}^{2k}\left|Cov\left(Y_i,Y_j\right)\right|\\ \le Ck,\end{array}$$

and

$$\begin{array}{ll}\hfill Var\left({\tilde{S}}_{2k}\right)& =Var\left(\sum _{i=1}^{2k}{Y}_i\right)\\ =\sum _{i=1}^{2k}E{Y}_i^2+2\sum _{i=1}^{2k-1}\sum _{j=i+1}^{2k}\left|Cov\left(Y_i,Y_j\right)\right|\\ =2k+2\sum _{i=1}^{2k-1}\sum _{j=2}^{2k-i+1}Cov\left(Y_i,Y_j\right)\\ \le Ck.\end{array}$$

Since f is a bounded Lipschitz function, we have

$$\begin{array}{l}\left|Cov\left(f\left(\frac{S_{k,k}}{{\sigma }_k}\right),f\left(\frac{S_{l,l}}{{\sigma }_l}\right)-f\left(\frac{S_{l,l}-S_{2k,2k}-b_{2k+1,l}{\tilde{S}}_{2k}}{{\sigma }_l}\right)\right)\right|\\ \le CE\frac{\left|S_{2k,2k}+b_{2k+1,l}{\tilde{S}}_{2k}\right|}{{\sigma }_l}\\ \le C\frac{{\left(Var\left(S_{2k,2k}\right)\right)}^{\frac{1}{2}}}{{\sigma }_l}+C\frac{b_{2k+1,l}{\left(Var\left({\tilde{S}}_{2k}\right)\right)}^{\frac{1}{2}}}{{\sigma }_l}\\ \le C{\left(\frac{k}{l}\right)}^{\frac{1}{2}}+C{\left(\frac{k}{l}\right)}^{\frac{1}{2}}\text{log}\frac{1}{2k}\\ \le C{\left(\frac{k}{l}\right)}^{\epsilon },\end{array}$$

where $0<\epsilon <\frac{1}{2}$. Hence if l > 2k, we have

$$\left|E{\xi }_k{\xi }_l\right|\le C\left[{\rho }^{-}\left(k\right)+{\left(\frac{k}{l}\right)}^{\epsilon }\right].$$

Thus

$$\begin{array}{ll}\hfill I_2& \le \frac{C}{{\text{log}}^{2}n}\sum _{l=2}^n\sum _{k=1}^{l-1}\frac{1}{k^{1-\epsilon }{l}^{1+\epsilon }}+\frac{C}{{\text{log}}^{2}n}\sum _{l=2}^n\frac{1}{l}\sum _{k=1}^{l-1}\frac{{\rho }^{-}\left(k\right)}{k}\\ \le \frac{C}{{\text{log}}^{2}n}\sum _{l=2}^n\frac{1}{l^{1+\epsilon }}\frac{{\left(l-1\right)}^{\epsilon }}{\epsilon }+\frac{C}{{\text{log}}^{2}n}\sum _{l=2}^n\frac{1}{l}\sum _{k=1}^n\frac{{\text{log}}^{-\delta }k}{k}\\ \le \frac{C}{{\text{log}}^{2}n}\sum _{l=2}^n\frac{1}{l}+\frac{C}{{\text{log}}^{2}n}\sum _{l=2}^n\frac{1}{l}\sum _{k=1}^n\frac{{\text{log}}^{-\delta }k}{k}\\ \le C{\text{log}}^{-1}n.\end{array}$$

(3.8)

Associated with (3.5), (3.6), and (3.8), we have

$$E{T}_n^2\le C{\text{log}}^{-1}n.$$

(3.9)

To prove (3.4), let $n_k=e^{k^{\tau }}$, where τ > 1. From (3.9), we have

$$\sum _{k=1}^{\infty }E{T}_{n_k}^2\le C\sum _{k=1}^{\infty }{\text{log}}^{-1}{n}_k=C\sum _{k=1}^{\infty }\frac{1}{k^{\tau }}<\infty .$$

Thus ∀ε > 0, we have

$$\sum _{k=1}^{\infty }P\left\{\left|T_{n_k}\right|\ge \epsilon \right\}\le \sum _{k=1}^{\infty }\frac{E{T}_{n_k}^2}{{\epsilon }^2}<\infty .$$

By Borel-Cantelli lemma, we have

$$T_{n_k\to 0}a.s.ask\to \infty .$$

Note that

$$\frac{\text{log}{n}_{k+1}}{\text{log}{n}_k}=\frac{{\left(k+1\right)}^{\tau }}{k^{\tau }}\to 1ask\to \infty .$$

For every n, there exist n _k and n_k+1satisfying n _k < n ≤ n_k+1, we have

$$\begin{array}{ll}\hfill \left|T_n\right|& \le \frac{1}{\text{log}{n}_k}\left|\sum _{i=1}^{n_k}\frac{{\xi }_i}{i}\right|+\frac{1}{\text{log}{n}_k}\sum _{i=n_k}^{n_{k+1}}\frac{\left|{\xi }_i\right|}{i}\\ \le \left|T_{n_k}\right|+C\left(\frac{\text{log}{n}_{k+1}}{\text{log}{n}_k}-1\right)\to 0a.s.asn\to \infty ,\end{array}$$

(3.4) is completed, so the proof of Theorem 1.1 is completed.

Proof of Theorem 1.2 Let $C_i=\frac{S_i}{{\mu }_i}$, we have

$$\frac{1}{\gamma {\sigma }_k}\sum _{i=1}^k\left(C_i-1\right)=\frac{1}{\gamma {\sigma }_k}\sum _{i=1}^k\left(\frac{S_i}{\mu i}-1\right)=\frac{1}{{\sigma }_k}\sum _{i=1}^k{b}_{i,k}{Y}_i=\frac{S_{k,k}}{{\sigma }_k}.$$

Hence (1.1) is equivalent to

$$\forall x\underset{n\to \infty }{\text{lim}}\frac{1}{\text{log}n}\sum _{k=1}^n\frac{1}{k}I\left\{\frac{1}{\gamma {\sigma }_k}\sum _{i=1}^k\left(C_i-1\right)\le x\right\}=\Phi \left(x\right)a.s.$$

(3.10)

On the other hand, to prove (1.2), it suffices to show that

$$\forall x\underset{n\to \infty }{\text{lim}}\frac{1}{\text{log}n}\sum _{k=1}^n\frac{1}{k}I\left\{\frac{1}{\gamma {\sigma }_k}\sum _{i=1}^k\text{log}{C}_i\le x\right\}=\Phi \left(x\right)a.s.$$

(3.11)

By Lemma 2.7, for enough large i, for some $\frac{4}{3}<p<2$ we have

$$\left|C_i-1\right|=\left|\frac{S_i}{\mu i}-1\right|\le C{i}^{\frac{1}{p}-1}a.s.$$

It is easy to know that log(1+ x) = x + O(x²) for $\left|x\right|<\frac{1}{2}$, thus

$$\left|\sum _{k=1}^n\text{log}{C}_k-\sum _{k=1}^n\left(C_k-1\right)\right|\le C\sum _{k=1}^n{\left(C_k-1\right)}^2\le C\sum _{k=1}^n{k}^{\frac{2}{p}-2}\le C{n}^{\frac{2}{p}-1}a.s.,$$

and

$$\sum _{k=1}^n\left(C_k-1\right)-C{n}^{\frac{2}{p}-1}\le \sum _{k=1}^n\text{log}{C}_k\le \sum _{k=1}^n\left(C_k-1\right)+C{n}^{\frac{2}{p}-1}a.s.$$

Hence for arbitrary small ε > 0, there is n₀ = n₀(ω, ε), such that for every n > n₀ and arbitrary x,

$$I\left\{\frac{1}{\gamma {\sigma }_k}\sum _{i=1}^k\left(C_i-1\right)\le x-\epsilon \right\}\le I\left\{\frac{1}{\gamma {\sigma }_k}\sum _{i=1}^k\text{log}{C}_i\le x\right\}\le I\left\{\frac{1}{\gamma {\sigma }_k}\sum _{i=1}^k\left(C_i-1\right)\le x+\epsilon \right\},$$

so by (3.10), we know that (3.11) is true, and (3.11) is equivalent to (1.2), thus the proof of Theorem 1.2 is complete.