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
1. Introduction and Main Result
Gut and Spătaru  proved its precise asymptotics as follows.
Theorem 1 A.
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.
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.
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.
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.
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