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

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.

Let be a sequence of i.i.d. positive random variables with and . Denote the coefficient of variation. Then for any real

(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.

Let be a sequence of identically distributed positive strongly mixing random variable with and , , . Denote by the coefficient of variation, and . Assume

(1.2)

Then for any real

(1.3)

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.

Let be a sequence of identically distributed positive strongly mixing random variable with and , and as mentioned above. Denote by the coefficient of variation, and . Assume that

(1.4)

(1.5)

(1.6)

(1.7)

Then for any real

(1.8)

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.

Under the conditions of Theorem 1.3, for any real

(1.9)

where 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]).

Let be a sequence of strongly mixing random variables with zero mean, and let be a triangular array of real numbers. Assume that

(2.1)

If for a certain is uniformly integrable, ,

(2.2)

then

(2.3)

Lemma 2.2 (see [9]).

Let ; then

(2.4)

where as as .

Lemma 2.3 (see [8]).

Let be a strongly mixing sequence of random variables such that for a certain and every . Then there is a numerical constant depending only on such that for every one has

(2.5)

where .

Lemma 2.4 (see [9]).

Let be a sequence of random variables, uniformly bounded below and with finite variances, and let be a sequence of positive number. Let for and . Assume that

(2.6)

as . If for some , and all

(2.7)

then

(2.8)

Lemma 2.5 (see [10]).

Let be a strongly mixing sequence of random variables with zero mean and for a certain . Assume that (1.5) and (1.6) hold. Then

(2.9)

2.2. Proof of Theorem 1.4

From the definition of strongly mixing we know that remain to be a sequence of identically distributed strongly mixing random variable with zero mean and unit variance. Let ; note that

(2.10)

and via (1.7) we have

(2.11)

From the definition of and (1.4) we have that is uniformly integrable; note that

(2.12)

and applying (1.5)

(2.13)

Consequently using Lemma 2.1, we can obtain

(2.14)

which is equivalent to

(2.15)

for any bounded Lipschitz-continuous function ; applying Toeplitz Lemma

(2.16)

We notice that (1.9) is equivalent to

(2.17)

for all bounded Lipschitz continuous ; it therefore remains to prove that

(2.18)

Let ,

(2.19)

From Lemma 2.2, we obtain for some constant

(2.20)

Using (2.20) and property of , we have

(2.21)

We estimate now . For ,

(2.22)

Notice that

(2.23)

and the properties of strongly mixing sequence imply

(2.24)

Applying Lemma 2.3 and (2.10),

(2.25)

Consequently, via the properties of , the Jensen inequality, and (1.7),

(2.26)

where . Hence for we have

(2.27)

Consequently, we conclude from the above inequalities that

(2.28)

Applying (1.5) and Lemma 2.2 we can obtain for any

(2.29)

Notice that

(2.30)

(2.31)

Let , then

(2.32)

By (2.21), (2.29), (2.31), and (2.32), for some such that

(2.33)

applying Lemma 2.4, we have

(2.34)

2.3. Proof of Theorem 1.3

Let ; we have

(2.35)

We see that (1.9) is equivalent to

(2.36)

Note that in order to prove (1.8) it is sufficient to show that

(2.37)

From Lemma 2.5, for sufficiently large , we have

(2.38)

Since for , thus

(2.39)

Hence for any and for sufficiently large , we have

(2.40)

and thus (2.36) implies (2.37).