- Research Article
- Open Access
A General Law of Complete Moment Convergence for Self-Normalized Sums
© Qing-pei Zang. 2010
- Received: 9 March 2010
- Accepted: 11 April 2010
- Published: 17 May 2010
Let be a sequence of independent and identically distributed (i.i.d.) random variables, and is in the domain of the normal law and . In this paper, we obtain a general law of complete moment convergence for self-normalized sums.
- Convergence Rate
- Asymptotic Normality
- Precise Rate
- Iterate Logarithm
- Strong Approximation
if and only if and Later, Chen  and Gut and Spătaru  both studied the precise asymptotics of the infinite sums as Moreover, Gut and Spătaru [9, 10] studied the precise asymptotics of the law of the iterated logarithm and the precise asymptotics for multidimensionally indexed random variables. Lanzinger and Stadtmüller , Spătaru [12, 13], and Huang and Zhang  obtained the precise rates in some different cases. While, Chow  discussed the complete moment convergence of i.i.d. random variables. He got the following result.
Thus, the complete moment convergence rates can reflect the convergence rates more directly than exact probability convergence rates.
For the investigation of complete moment convergence, some authors have researched it in different directions. For example, Jiang and Zhang  derived the precise asymptotics in the law of the iterated logarithm for the moment convergence of i.i.d. random variables by using the strong approximation method.
Liu and Lin  introduced a new kind of complete moment convergence, Li  got precise asymptotics in complete moment convergence of moving-average processes, Zang and Fu  obtained precise asymptotics in complete moment convergence of the associated counting process, and Fu  also investigated asymptotics for the moment convergence of U-Statistics in LIL.
In the recent years, the limit theorems for self-normalized sum or, equivalently, Student -statistics , have attracted more and more attention. Bentkus and Götze  obtained Berry-Esseen inequalities for self-normalized sums. Wang and Jing  derived exponential nonuniform Berry-Esseen bound. Hu et al.  achieved cramér type moderate deviations for the maximum of self-normalized sums. Giné et al.  established asymptotic normality of self-normalized sums as follows.
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.
Meanwhile, Shao  showed a self-normalized large deviation result for without any moment conditions.
Inspired by the above results, the purpose of this paper is to study a general law of complete moment convergence for self-normalized sums. Our main result is as follows.
In Theorem 1.1, the condition is mild. For example, with some suitable conditions of , and and some others all satisfy this condition.
Obviously, our main result is the generalization of i.i.d. random variables which have the finiteness of the second moments.
As examples, in Theorem 1.1, we can obtain some corollaries by choosing different and as follows.
In this section, let for and is the inverse function of . Here and in the sequel, will denote positive constants, possibly varying from place to place. Theorem 1.1 will be proved via the following propositions.
Here and in the sequel, denotes the standard normal random variable.
then, by (2.2) and letting we complete the proof of this proposition.
The proof is completed.
Then, this proposition holds.
Combining (2.20), (2.21), and (2.22), the proposition is proved.
Theorem 1.1 now follows from the above propositions using the triangle inequality.
The author would like to thank an associate editor and the reviewer for their pointing out some serious problems in previous version. Thanks are also paid to the author's supervisor professor Zhengyan Lin of Zhejiang University in China and Dr. Keang Fu of Zhejiang Gongshang University in China for their help.
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