• Research Article
• Open Access

# Asymptotics for the Moment Convergence of -Statistics in LIL

Journal of Inequalities and Applications20102010:350517

https://doi.org/10.1155/2010/350517

• Received: 18 September 2009
• Accepted: 18 January 2010
• Published:

## Abstract

Let be a -statistic based on a symmetric kernel and i.i.d. samples . In this paper, the exact moment convergence rates in the law of the iterated logarithm and the law of the logarithm of are obtained, which extend previous results concerning partial sums.

## Keywords

• Convergence Rate
• Limit Theorem
• Error Variance
• Central Limit
• Central Limit Theorem

## 1. Introduction and Main Result

Let be a real-valued Borel measurable function, symmetric in its arguments. Define a -statistic based on an independent and identically distributed (i.i.d.) sequence and kernel function as follows:

This class of -statistics was introduced by Hoeffding  and Halmos  in the 1940s, and we have witnessed a rapid development in asymptotic theory of -statistics since then (see Koroljuk and Borovskich  and Serfling  for more details).

It is well known that, initiating from the work of Gut and Spătaru , many authors devoted themselves to the research of precise asymptotics. Recently, Zhou et al.  studied the precise asymptotics of a special kind of statistics, which includes the U-statistics, Von-Mises statistics, linear processes, moving average processes, error variance estimates in linear models and power sums. One of their main results is as follows, which reflects the exact probability convergence rate in the law of the iterated logarithm.

Theorem 1 A.

Let be a sequence of i.i.d. random variables with mean zero and variance one. Let be a random function or statistic satisfying where If then for any where is the Gamma function and Since Theorem A requires a strong condition, that is, Yan and Su  investigated the precise asymptotics of -statistics under minimal conditions and got the following result.

Theorem 1 B.

Let be a -statistic given by (1.1). Suppose that for some , , and where and Then for any On the other hand, for the i.i.d. sequence it is noted that Chow  first introduced the well-known complete moment convergence and gave the result as follows.

Theorem 1 C.

Suppose that For , and if , then for any where Inspired by them, in this paper, we aim to establish a moment version of Theorem B for -statistics. Our main result reads as follows.

Theorem 1.1.

Let be a -statistic given by (1.1). Suppose that and Then for any where is a normal random variable with mean zero and variance .

Remark.

Here we consider the moment convergence rates of U-statistic in the law of the iterated logarithm, extending the results of Zhou et al.  and Yan and Su  for exact probability convergence rates and reflecting the convergence rates of the law of the iterated logarithm more directly.

By some modifications, we can get the following result easily.

Theorem 1.3.

Under the assumptions of Theorem 1.1, One has that for and Remark.

Note that in our theorem, we assume which is stronger than the condition imposed by Yan and Su , and required only to use a moment bound of Chen  given in Lemma 2.1. However, the assumption in Yan and Su  is weakened.

## 2. Proof of Theorem 1.1

Note that readily implies Thus without loss of generality, assume In the sequel, let denote a positive constant whose value possibly varies from place to place and the notation of means the integer part of We first introduce some useful lemmas, which are known as the moment inequality of -statistics and the Toeplitz lemma, respectively.

Lemma (Chen ).

Let be given by (1.1). Suppose that and for Then there exists a constant depending only on such that

Lemma (Stout ).

Let be a matrix of real numbers and a sequence of real numbers. Let as Then

In what follows, for and we set The proof is very much modeled for proving results in the area of precise asymptotics, and hence Theorem 1.1 follows immediately by applying the following propositions.

Proposition 2.3.

For any , one has

where is defined as above.

Proof.

Proposition 2.4.

For one has

Proof.

Set . Then, from the central limit theorem for -statistics (cf. Koroljuk and Borovskich ), it follows that as Note that
Thus, for by applying Lemma 2.2, we have
As for coupled with Markov's inequality and Lemma 2.1 with , then an application of Lemma 2.2 provides

Hence (2.6) holds true.

Proposition 2.5.

For and , one has uniformly

Proof.

Note that for large enough,

when , uniformly for Proposition 2.6.

Proof.

Notice that by virtue of Lemma 2.1 with , it follows that

Proof of Theorem 1.1.

Theorem 1.1 follows from Propositions 2.3–2.6 by using the triangle inequality immediately.

## 3. Proof of Theorem 1.3

By some simple modifications, Theorem 1.3 can be got similarly. For completeness, we state the similar Propositions 3.1–3.4 in the following without details.

Proposition 3.1.

For and one has

Proposition 3.2.

For and one has

where Proposition 3.3.

For and one has

Proposition 3.4.

For and one has

## Declarations

### Acknowledgments

This project is supported by National Natural Science Foundation of China (10901138), the Introduction Talent Foundation of Zhejiang Gongshang University (1020XJ200961), and the Research Grant of Zhejiang Gongshang University (X10-26).

## Authors’ Affiliations

(1)
School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, 310018, China

## References 