- Research Article
- Open Access

- Ke-Ang Fu
^{1}Email author

**2010**:350517

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

© Ke-Ang Fu. 2010

**Received:**18 September 2009**Accepted:**18 January 2010**Published:**2 February 2010

## Abstract

## 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 [1] and Halmos [2] in the 1940s, and we have witnessed a rapid development in asymptotic theory of -statistics since then (see Koroljuk and Borovskich [3] and Serfling [4] for more details).

It is well known that, initiating from the work of Gut and Spătaru [5], many authors devoted themselves to the research of precise asymptotics. Recently, Zhou et al. [6] 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.

where is the Gamma function and

Since Theorem A requires a strong condition, that is, Yan and Su [7] investigated the precise asymptotics of -statistics under minimal conditions and got the following result.

Theorem 1 B.

On the other hand, for the i.i.d. sequence it is noted that Chow [8] first introduced the well-known complete moment convergence and gave the result as follows.

Theorem 1 C.

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.

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. [6] and Yan and Su [7] 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.

Remark.

Note that in our theorem, we assume which is stronger than the condition imposed by Yan and Su [7], and required only to use a moment bound of Chen [9] given in Lemma 2.1. However, the assumption in Yan and Su [7] 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 [9]).

Lemma (Stout [10]).

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.

Proof.

Proposition 2.4.

Proof.

Hence (2.6) holds true.

Proposition 2.5.

Proof.

Proposition 2.6.

Proof.

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.

Proposition 3.2.

Proposition 3.3.

Proposition 3.4.

## 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

## References

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## Copyright

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