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

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## Abstract

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

### Keywords

Real Number Recent Decade Limit Theorem Central Limit Central Limit Theorem## 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 Open image in new window , Open image in new window is the indicator function, Open image in new window is the distribution function of the random variable Open image in new window , and Open image in new window 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 Open image in new window 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 Open image in new window with Open image in new window and Open image in new window [7]. Clearly, the larger the weight sequence Open image in new window is, the stronger is the result (1.3).

In the following sections, let Open image in new window , Open image in new window , " Open image in new window " denote the inequality " Open image in new window " 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 Open image in new window and Open image in new window for Open image in new window with Open image in new window if Open image in new window . Open image in new window , Open image in new window , Open image in new window and Open image in new window .

In this setting we establish an ASCLT for the triangular array Open image in new window .

Theorem 1.4.

where Open image in new window is the standard normal distribution function.

## 2. The Proofs

### 2.1. Lemmas

To prove theorems, we need the following lemmas.

Lemma 2.1 (see [8]).

Lemma 2.2 (see [9]).

where Open image in new window as Open image in new window as Open image in new window .

Lemma 2.3 (see [8]).

where Open image in new window .

Lemma 2.4 (see [9]).

Lemma 2.5 (see [10]).

### 2.2. Proof of Theorem 1.4

*Toeplitz Lemma*

*Jensen*inequality, and (1.7),

### 2.3. Proof of Theorem 1.3

and thus (2.36) implies (2.37).

## Notes

### Acknowledgment

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

### References

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