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.

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 [7] 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 [8] 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. [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.

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 [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.