# 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.

## 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

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## Copyright

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