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A General Law of Complete Moment Convergence for Self-Normalized Sums
Journal of Inequalities and Applications volume 2010, Article number: 760735 (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.
1. Introduction and Main Results
Let be a sequence of independent and identically distributed (i.i.d.) random variables and put
for . We have the famous result following, that is, the complete convergence, for and ,
if and only if and when For , the sufficiency was proved by Hsu and Robbins , and the necessity by Erdös [2, 3]. For the case , we refer to Spitzer , and one can refer to Baum and Katz  for the general result. Note that the sums obviously tend to infinity as Thus it is interesting to discuss the precise rate and limit the value of as , where and are the positive functions defined on . We call and weighted function and boundary function, respectively. The first result in this direction was due to Heyde , who proved that
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.
Let be a sequence of i.i.d. random variables with . Suppose that and Then for any , one has
An important observation is that
From (1.5), we obtain that the complete moment convergence implies the complete convergence, that is, under the conditions of Theorem A, result (1.4) implies that
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.
Let be a sequence of i.i.d. random variables with , , and . Set . Then for , one has
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.
On the other hand, the so-called self-normalized sum is of the form . Using this notation we can write the classical Student -statistics as
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.
Let be a sequence of i.i.d. random variables with . Then for any ,
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.
Let be a sequence of positive numbers with and as . If and is slowly varying as then
In view of this theorem, and by applying s to it, one can obtain that for large enough and any , there exist and such that for . In particular, for , there exists such that
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.
Suppose is in the domain of attraction of the normal law and . Assume that is differentiable on the interval , which is strictly increasing to , and differentiable function is nonnegative. Suppose that is monotone and . If is monotone nondecreasing, one assumes that Then, for , one has
In Theorem 1.1, the condition is mild. For example, with some suitable conditions of , and and some others all satisfy this condition.
If , by the strong law of large numbers, we have Then, we can easily obtain the following result:
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.
Let , where , one has
Let , where , one has
Let , where , one has
2. Proof of Theorem 1.1
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.
Via the change of variable, for arbitrary , we have
Thus, if is monotone nonincreasing, then is nonincreasing. Hence
then, by (2.2), the proposition holds. If is nondecreasing, then by , for any there exists such that and for Thus we have
then, by (2.2) and letting we complete the proof of this proposition.
It is easy to see, from (1.9), that , as Observe that
Thus for , it is easy to see that
Now we are in a position to estimate . From (1.11) and by Markov's inequality, we have
For , by Markov's inequality and (1.11), we have
From Cauchy inequality, it follows that
Denote . Note that . Then, since the weighted average of a sequence that converges to 0 also converges to 0, it follows that, for any ,
The proof is completed.
By the similar argument in Proposition 2.1, it follows that
Then, this proposition holds.
By the similar argument in Proposition 2.1, it follows that
For , by (1.11), we have
For , using (1.11) again and noticing that , we have
By noting (2.12), it is easily seen that
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.
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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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Zang, Qp. A General Law of Complete Moment Convergence for Self-Normalized Sums. J Inequal Appl 2010, 760735 (2010). https://doi.org/10.1155/2010/760735
- Convergence Rate
- Asymptotic Normality
- Precise Rate
- Iterate Logarithm
- Strong Approximation