# Almost Sure Central Limit Theorem for Product of Partial Sums of Strongly Mixing Random Variables

- Daxiang Ye
^{1}Email author and - Qunying Wu
^{1}

**2011**:576301

https://doi.org/10.1155/2011/576301

© Daxiang Ye and Qunying Wu. 2011

**Received: **19 September 2010

**Accepted: **26 January 2011

**Published: **15 February 2011

## Abstract

We give here an almost sure central limit theorem for product of sums of strongly mixing positive random variables.

## Keywords

## 1. Introduction and Results

In recent decades, there has been a lot of work on the almost sure central limit theorem (ASCLT), we can refer to Brosamler [1], Schatte [2], Lacey and Philipp [3], and Peligrad and Shao [4].

Khurelbaatar and Rempala [5] gave an ASCLT for product of partial sums of i.i.d. random variables as follows.

Theorem 1.1.

where , is the indicator function, is the distribution function of the random variable , and is a standard normal variable.

Recently, Jin [6] had proved that (1.1) holds under appropriate conditions for strongly mixing positive random variables and gave an ASCLT for product of partial sums of strongly mixing as follows.

Theorem 1.2.

The sequence in (1.3) is called weight. Under the conditions of Theorem 1.2, it is easy to see that (1.3) holds for every sequence with and [7]. Clearly, the larger the weight sequence is, the stronger is the result (1.3).

In the following sections, let , , " " denote the inequality " " up to some universal constant.

We first give an ASCLT for strongly mixing positive random variables.

Theorem 1.3.

In order to prove Theorem 1.3 we first establish ASCLT for certain triangular arrays of random variables. In the sequel we shall use the following notation. Let and for with if . , , and .

In this setting we establish an ASCLT for the triangular array .

Theorem 1.4.

## 2. The Proofs

### 2.1. Lemmas

To prove theorems, we need the following lemmas.

Lemma 2.1 (see [8]).

Lemma 2.2 (see [9]).

Lemma 2.3 (see [8]).

Lemma 2.4 (see [9]).

Lemma 2.5 (see [10]).

### 2.2. Proof of Theorem 1.4

### 2.3. Proof of Theorem 1.3

and thus (2.36) implies (2.37).

## Declarations

### Acknowledgment

This work is supported by the National Natural Science Foundation of China (11061012), Innovation Project of Guangxi Graduate Education (200910596020M29).

## Authors’ Affiliations

## References

- Brosamler GA:
**An almost everywhere central limit theorem.***Mathematical Proceedings of the Cambridge Philosophical Society*1988,**104**(3):561–574. 10.1017/S0305004100065750MATHMathSciNetView ArticleGoogle Scholar - Schatte P:
**On strong versions of the central limit theorem.***Mathematische Nachrichten*1988,**137:**249–256. 10.1002/mana.19881370117MATHMathSciNetView ArticleGoogle Scholar - Lacey MT, Philipp W:
**A note on the almost sure central limit theorem.***Statistics & Probability Letters*1990,**9**(3):201–205. 10.1016/0167-7152(90)90056-DMATHMathSciNetView ArticleGoogle Scholar - Peligrad M, Shao QM:
**A note on the almost sure central limit theorem for weakly dependent random variables.***Statistics & Probability Letters*1995,**22**(2):131–136. 10.1016/0167-7152(94)00059-HMATHMathSciNetView ArticleGoogle Scholar - Khurelbaatar G, Rempala G:
**A note on the almost sure central limit theorem for the product of partial sums.***Applied Mathematics Letters*2004,**19:**191–196.View ArticleGoogle Scholar - Jin JS:
**An almost sure central limit theorem for the product of partial sums of strongly missing random variables.***Journal of Zhejiang University*2007,**34**(1):24–27.MATHMathSciNetGoogle Scholar - Berkes I, Csáki E:
**A universal result in almost sure central limit theory.***Stochastic Processes and Their Applications*2001,**94**(1):105–134. 10.1016/S0304-4149(01)00078-3MATHMathSciNetView ArticleGoogle Scholar - Peligrad M, Utev S:
**Central limit theorem for linear processes.***The Annals of Probability*1997,**25**(1):443–456.MATHMathSciNetView ArticleGoogle Scholar - Jonsson F:
*Almost Sure Central Limit Theory*. Uppsala University: Department of Mathematics; 2007.Google Scholar - Chuan-Rong L, Zheng-Yan L:
*Limit Theory for Mixing Dependent Random Variabiles*. Science Press, Beijing, China; 1997.Google Scholar

## Copyright

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