- Research Article
- Open Access

# A Limit Theorem for the Moment of Self-Normalized Sums

- Qing-pei Zang
^{1}Email author

**2009**:957056

https://doi.org/10.1155/2009/957056

© Qing-pei Zang. 2009

**Received:**25 December 2008**Accepted:**18 June 2009**Published:**14 July 2009

## Abstract

Let
be a sequence of independent and identically distributed (*i.i.d.*) random variables and
is in the domain of attraction of the normal law and
. For
, we prove the precise asymptotics in Davis law of large numbers for

## Keywords

- Limit Theorem
- Triangle Inequality
- Significant Development
- Asymptotic Normality
- Strong Approximation

## 1. Introduction and Main Result

if and only if and .

Gut and Spătaru [2] proved its precise asymptotics as follows.

Theorem 1 A.

where stands for the absolute moment of the standard normal distribution.

It is well known that, for random variables, Chow [3] discussed the complete moment convergence, and got the following result.

Theorem 1 B.

On the other hand, the past decade has witnessed a significant development on the limit theorems for the so-called self-normalized sum . Bentkus and Götze [4] obtained Berry-Esseen inequalities for self-normalized sums. Wang and Jing [5] derived exponential nonuniform Berry-Esseen bound. Giné et al. [6], established asymptotic normality of self-normalized sums.

Theorem 1 C.

holds, if and only if is in the domain of attraction of the normal law, where is the distribution function of the standard normal random variable.

Shao [7] showed a self-normalization large deviation result for without any moment conditions.

Theorem 1 D.

Since then, many subsequent developments of self-normalized sums have been obtained. For example, Csörgő et al. [8] have established Darling-Erdös theorem for self-normalized sums, and they [9] have also obtained Donsker's theorem for self-normalized partial sums processes.

Inspired by the above results, in this note we study the precise asymptotics in Davis law of large numbers for the moment of self-normalized sums. Our main result is as follows.

Theorem 1.1.

here and in the sequel, is the standard normal random variable.

Remark 1.2.

Remark 1.3.

As is well known, the strong approximation method is taken in order to obtain such an analogous result, however, this method is not applicable here.

## 2. Proof of Theorem 1.1

In this section, we set for and . Here and in the sequel, will denote positive constants, possibly varying from place to place, and means the largest integer . The proof of Theorem 1.1 is based on the following propositions.

Proposition 2.1.

Proof.

Proposition 2.2.

Proof.

The proof is completed.

Proposition 2.3.

Proof.

So this proposition is proved now.

Proposition 2.4.

Proof.

Combining (2.18), (2.19), and (2.20), the proposition is proved.

Our main result follows from the propositions using the triangle inequality.

## Declarations

### Acknowledgments

The author thanks the referees for pointing out some errors in a previous version, as well as for several comments that have led to improvements in this work. Thanks are also due to Doctor Ke-ang Fu of Zhejiang University in china for his valuable suggestion in the preparation of this paper.

## Authors’ Affiliations

## References

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## Copyright

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.