- Research Article
- Open Access
A Limit Theorem for the Moment of Self-Normalized Sums
© Qing-pei Zang. 2009
- Received: 25 December 2008
- Accepted: 18 June 2009
- Published: 14 July 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
- Limit Theorem
- Triangle Inequality
- Significant Development
- Asymptotic Normality
- Strong Approximation
if and only if and .
Gut and Spătaru  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  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  obtained Berry-Esseen inequalities for self-normalized sums. Wang and Jing  derived exponential nonuniform Berry-Esseen bound. Giné et al. , 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  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.  have established Darling-Erdös theorem for self-normalized sums, and they  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.
here and in the sequel, is the standard normal random variable.
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.
The proof is completed.
So this proposition is proved now.
Combining (2.18), (2.19), and (2.20), the proposition is proved.
Our main result follows from the propositions using the triangle inequality.
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.
- Davis JA: Convergence rates for probabilities of moderate deviations. Annals of Mathematical Statistics 1968, 39: 2016–2028. 10.1214/aoms/1177698029MathSciNetView ArticleMATHGoogle Scholar
- Gut A, Spătaru A: Precise asymptotics in the law of the iterated logarithm. The Annals of Probability 2000,28(4):1870–1883. 10.1214/aop/1019160511MathSciNetView ArticleMATHGoogle Scholar
- Chow 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.MathSciNetMATHGoogle Scholar
- Bentkus V, Götze F: The Berry-Esseen bound for Student's statistic. The Annals of Probability 1996,24(1):491–503.MathSciNetView ArticleMATHGoogle Scholar
- 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/1022677562MathSciNetView ArticleMATHGoogle Scholar
- Giné E, Götze F, Mason DM: When is the student -statistic asymptotically standard normal? The Annals of Probability 1997,25(3):1514–1531.MathSciNetView ArticleMATHGoogle Scholar
- Shao Q-M: Self-normalized large deviations. The Annals of Probability 1997,25(1):285–328.MathSciNetView ArticleMATHGoogle Scholar
- Csörgő M, Szyszkowicz B, Wang Q: Darling-Erdős theorem for self-normalized sums. The Annals of Probability 2003,31(2):676–692.MathSciNetView ArticleMATHGoogle Scholar
- 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/1055425777MathSciNetView ArticleMATHGoogle Scholar
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.