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Journal of Applied Mathematics
Volume 2012, Article ID 329391, 13 pages
http://dx.doi.org/10.1155/2012/329391
Research Article

Almost Sure Central Limit Theory for Self-Normalized Products of Sums of Partial Sums

1College of Science, Guilin University of Technology, Guilin 541004, China
2Guangxi Key Laboratory of Spatial Information and Geomatics, Guilin University of Technology, Guilin 541004, China

Received 18 January 2012; Accepted 7 February 2012

Academic Editor: Naseer Shahzad

Copyright © 2012 Qunying Wu. 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.

Abstract

Let be a sequence of independent and identically distributed random variables in the domain of attraction of a normal law. An almost sure limit theorem for the self-normalized products of sums of partial sums is established.

1. Introduction

Let be a sequence of independent and identically distributed (i.i.d.) positive random variables with a nondegenerate distribution function and . For each , the symbol denotes self-normalized partial sums, where and . We say that the random variable belongs to the domain of attraction of the normal law if there exist constants , such that here and in the sequel is a standard normal random variable. We say that satisfies the central limit theorem (CLT).

It is known that (1.1) holds if and only if In contrast to the well-known classical central limit theorem, Gine et al. [1] obtained the following self-normalized version of the central limit theorem: as if and only if (1.2) holds.

Brosamler [2] and Schatte [3] obtained the following almost sure central limit theorem (ASCLT): let be i.i.d. random variables with mean 0, variance , and partial sums . Then with and ; here and in the sequel denotes an indicator function, and is the standard normal distribution function. Some ASCLT results for partial sums were obtained by Lacey and Philipp [4], Ibragimov and Lifshits [5], Miao [6], Berkes and Csáki [7], Hörmann [8], Wu [9, 10], and Ye and Wu [11]. Huang and Pang [12] and Zhang and Yang [13] obtained ASCLT results for self-normalized version. However, ASCLT results for self-normalized products of sums of partial sums have not been reported yet.

Under mild moment conditions, ASCLT follows from the ordinary CLT, but in general the validity of ASCLT is a delicate question of a totally different character as CLT. The difference between CLT and ASCLT lies in the weight in ASCLT. The terminology of summation procedures (see e.g., Chandrasekharan and Minakshisundaram [14], page 35) shows that the larger the weight sequence in (1.3) is, the stronger the relation becomes. By this argument, one should also expect to get stronger results if we use larger weights. And it would be of considerable interest to determine the optimal weights.

On the other hand, by the Theorem  1 of Schatte [3], (1.3) fails for weight . The optimal weight sequence remains unknown.

The purpose of this paper is to study and establish the ASCLT for self-normalized products of sums of partial sums of random variables in the domain of attraction of the normal law, we will show that the ASCLT holds under a fairly general growth condition on , .

In the following, we assume that is a sequence of i.i.d. positive random variables in the domain of attraction of the normal law with and define . Let , and for . denotes the indicator function of set , and denotes . The symbol stands for a generic positive constant, which may differ from one place to another. Let

By the definition of , we have and for any . It implies that

Our theorem is formulated in a more general setting.

Theorem 1.1. Let be a sequence of i.i.d. positive random variables in the domain of attraction of the normal law with . Suppose
For , set Then for any , where is the distribution function of the random variable .

By the terminology of summation procedures, we have the following corollary.

Corollary 1.2. Theorem 1.1 remains valid if we replace the weight sequence by such that , .

Remark 1.3. If , then is in the domain of attraction of the normal law and , thus (1.6) holds. Therefore, the class of random variables in Theorems 1.1 is of very broad range.

Remark 1.4. Whether Theorem 1.1 holds for remains open.

2. Proofs

Furthermore, the following four lemmas will be useful in the proof, and the first is due to Csörgo et al. [15].

Lemma 2.1. Let be a random variable with , and denote . The following statements are equivalent: (i) is a slowly varying function at ;(ii) is in the domain of attraction of the normal law;(iii);(iv);(v) for .

Lemma 2.2. Let be a sequence of uniformly bounded random variables. If there exist constants and such that then where and are defined by (1.7).

Proof. We can easily apply the similar arguments of (2.1) in Wu [16] to get Lemma 2.2, and we omit the details here.

The following Lemma 2.3 can be directly verified.

Lemma 2.3. (i) ;
(ii); (iii); (iv).

For every , let

Lemma 2.4. Suppose that the assumptions of Theorem 1.1 hold. Then where and are defined by (1.7) and is a nonnegative, bounded Lipschitz function.

Proof. By , Lemma 2.1(iv), we have Thus, by (1.5) and Lemma 2.3 (iv), By (1.5) and Lemma 2.3 (i), Thus by combining Lemma 2.3 (iv), (1.6), and (2.8), Lindeberg condition hold.
Hence, it follows that This implies that for any , which is a nonnegative, bounded Lipschitz function, Hence, we obtain from the Toeplitz lemma.
On the other hand, note that (2.4) is equivalent to from Theorem  7.1 of Billingsley [17] and Section  2 of Peligrad and Shao [18]. Hence, to prove (2.4), it suffices to prove for any which is a nonnegative, bounded Lipschitz function.
Let
Clearly, there is a constant such that
For any , note that
For any , note that and are independent, is a nonnegative, bounded Lipschitz function, , , and . By the definition of , we get By Lemma 2.2, (2.15) holds.
Now we prove (2.5). Let It is known that for any sets and ; then for , by Lemma 2.1 (iii) and (1.5), we get Hence for , By Lemma 2.2, (2.5) holds.
Finally, we prove (2.6). Let For , By Lemma 2.2, (2.6) holds. This completes the proof of Lemma 2.4.

Proof of Theorem 1.1. Let ; then (1.8) is equivalent to where is the distribution function of the standard normal random variable .
Let , then . Using Marcinkiewicz-Zygmund strong large number law, we have Thus, Hence by for , for any , from , is a slowly varying function at , and .
Hence for almost every event and any , there exists such that for ,
Note that under condition ,
Thus, for any given , , combining (2.29), we have for Therefore, to prove (2.25), it suffices to prove for any and .
Firstly, we prove (2.32). Let and be a real function, such that for any given ,
By , Lemma 2.1 (iv) and (1.5), we have This, combining with (2.4), (2.36) and the arbitrariness of in (2.36), (2.32) holds.
By (2.5), (2.21) and the Toeplitz lemma, That is, (2.33) holds.
Now we prove (2.34). For any given , let be a nonnegative, bounded Lipschitz function such that
From , is i.i.d., , Lemma 2.1 (v), and (1.5), Therefore, combining (2.6) and the Toeplitz lemma, Hence, (2.34) holds. By similar methods used to prove (2.34), we can prove (2.35). This completes the proof of Theorem 1.1.

Acknowledgments

The author is very grateful to the referees and the Editors for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper. This paper is supported by the National Natural Science Foundation of China (11061012), a project supported by Program to Sponsor Teams for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning ([2011] 47), and the support program of Key Laboratory of Spatial Information and Geomatics (1103108-08).

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