## 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

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

## Abstract

## 1. Introduction and Main Result

## 2. Proof of Theorem 1.1

## References

## Acknowledgments

## Author information

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*Journal of Inequalities and Applications*
**volume 2009**, Article number: 957056 (2009)

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

Throughout this paper, we let be a sequence of random variables and is in the domain of attraction of the normal law and . Put

(1.1)

Also let Then by the well-known Davis laws of large numbers [1],

(1.2)

if and only if and .

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

Theorem 1 A.

Suppose that and Then for ,

(1.3)

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.

Let be a sequence of random variables with . Assume , , and Then for any ,

(1.4)

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.

Let be a sequence of random variables with . Then for any ,

(1.5)

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.

Let be a sequence of positive numbers with and as . If and is slowly varying as then

(1.6)

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.

Suppose is in the domain of attraction of the normal law and . Then, for and , one has

(1.7)

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

Remark 1.2.

If and , by the strong law of large numbers, we have Then, we can easily obtain the following result:

(1.8)

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.

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.

For , one has

(2.1)

Proof.

Via the change of variable , we have

(2.2)

Proposition 2.2.

For , one has

(2.3)

Proof.

Set Then, by (1.5), it is easy to see as Observe that

(2.4)

where

(2.5)

Thus for , it is easy to see

(2.6)

Now we are in a position to estimate . From (1.6), and by applying to it, we can obtain that for large enough and any , there exist C and b such that for . In particular, for , there exists such that

(2.7)

Hence, by Markov's inequality and (2.7), we have

(2.8)

For , by Markov's inequality and (2.7), we have

(2.9)

From Cauchy inequality, it follows that

(2.10)

Therefore

(2.11)

Denote , then, since the weighted average of a sequence that converges to 0 also converges to 0, it follows that, for any ,

(2.12)

The proof is completed.

Proposition 2.3.

For , one has

(2.13)

Proof.

Note that

(2.14)

So this proposition is proved now.

Proposition 2.4.

For , one has

(2.15)

Proof.

Note that

(2.16)

where

(2.17)

For , by (2.7), we have

(2.18)

For , using (2.7) again, we have

(2.19)

By noting that (2.10), it is easily seen that

(2.20)

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

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

Davis JA:

**Convergence rates for probabilities of moderate deviations.***Annals of Mathematical Statistics*1968,**39:**2016–2028. 10.1214/aoms/1177698029Gut A, Spătaru A:

**Precise asymptotics in the law of the iterated logarithm.***The Annals of Probability*2000,**28**(4):1870–1883. 10.1214/aop/1019160511Chow YS:

**On the rate of moment convergence of sample sums and extremes.***Bulletin of the Institute of Mathematics. Academia Sinica*1988,**16**(3):177–201.Bentkus V, Götze F:

**The Berry-Esseen bound for Student's statistic.***The Annals of Probability*1996,**24**(1):491–503.Wang Q, Jing B-Y:

**An exponential nonuniform Berry-Esseen bound for self-normalized sums.***The Annals of Probability*1999,**27**(4):2068–2088. 10.1214/aop/1022677562Giné E, Götze F, Mason DM:

**When is the student****-statistic asymptotically standard normal?***The Annals of Probability*1997,**25**(3):1514–1531.Shao Q-M:

**Self-normalized large deviations.***The Annals of Probability*1997,**25**(1):285–328.Csörgő M, Szyszkowicz B, Wang Q:

**Darling-Erdős theorem for self-normalized sums.***The Annals of Probability*2003,**31**(2):676–692.Csörgő M, Szyszkowicz B, Wang Q:

**Donsker's theorem for self-normalized partial sums processes.***The Annals of Probability*2003,**31**(3):1228–1240. 10.1214/aop/1055425777

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.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Zang, Qp. A Limit Theorem for the Moment of Self-Normalized Sums.
*J Inequal Appl* **2009**, 957056 (2009). https://doi.org/10.1155/2009/957056

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

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