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

Open Access
Research Article

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.

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

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.

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

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.

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

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.

Under the conditions of Theorem 1.3, for any real Open image in new window

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]).

Let Open image in new window be a sequence of strongly mixing random variables with zero mean, and let Open image in new window be a triangular array of real numbers. Assume that
If for a certain Open image in new window is uniformly integrable, Open image in new window ,

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]).

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

where Open image in new window .

Lemma 2.4 (see [9]).

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

Lemma 2.5 (see [10]).

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

2.2. Proof of Theorem 1.4

From the definition of strongly mixing we know that Open image in new window remain to be a sequence of identically distributed strongly mixing random variable with zero mean and unit variance. Let Open image in new window ; note that
and via (1.7) we have
From the definition of Open image in new window and (1.4) we have that Open image in new window is uniformly integrable; note that
and applying (1.5)
Consequently using Lemma 2.1, we can obtain
which is equivalent to
for any bounded Lipschitz-continuous function Open image in new window ; applying Toeplitz Lemma
We notice that (1.9) is equivalent to
for all bounded Lipschitz continuous Open image in new window ; it therefore remains to prove that
From Lemma 2.2, we obtain for some constant Open image in new window
Using (2.20) and property of Open image in new window , we have
Notice that
and the properties of strongly mixing sequence imply
Applying Lemma 2.3 and (2.10),
Consequently, via the properties of Open image in new window , the Jensen inequality, and (1.7),
Consequently, we conclude from the above inequalities that
Applying (1.5) and Lemma 2.2 we can obtain for any Open image in new window
Notice that
By (2.21), (2.29), (2.31), and (2.32), for some Open image in new window such that
applying Lemma 2.4, we have

2.3. Proof of Theorem 1.3

We see that (1.9) is equivalent to
Note that in order to prove (1.8) it is sufficient to show that
From Lemma 2.5, for sufficiently large Open image in new window , we have
Hence for any Open image in new window and for sufficiently large Open image in new window , we have

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

  1. 1.
    Brosamler GA: An almost everywhere central limit theorem. Mathematical Proceedings of the Cambridge Philosophical Society 1988,104(3):561–574. 10.1017/S0305004100065750MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Schatte P: On strong versions of the central limit theorem. Mathematische Nachrichten 1988, 137: 249–256. 10.1002/mana.19881370117MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    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-DMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    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-HMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    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.CrossRefGoogle Scholar
  6. 6.
    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
  7. 7.
    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)00078-3MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Peligrad M, Utev S: Central limit theorem for linear processes. The Annals of Probability 1997,25(1):443–456.MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Jonsson F: Almost Sure Central Limit Theory. Uppsala University: Department of Mathematics; 2007.Google Scholar
  10. 10.
    Chuan-Rong L, Zheng-Yan L: Limit Theory for Mixing Dependent Random Variabiles. Science Press, Beijing, China; 1997.Google Scholar

Copyright information

© Daxiang Ye and Qunying Wu. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.College of Science, Guilin University of TechnologyGuilinChina

Personalised recommendations