Asymptotics for the Moment Convergence of -Statistics in LIL

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:

(1.1)

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

(1.2)

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

(1.3)

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

(1.4)

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

(1.5)

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

(1.6)

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.

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]).

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

(2.1)

Lemma (Stout [10]).

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

(2.2)

imply that

(2.3)

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

(2.4)

where is defined as above.

Proof.

Notice that

(2.5)

Proposition 2.4.

For one has

(2.6)

Proof.

Set . Then, from the central limit theorem for -statistics (cf. Koroljuk and Borovskich [3]), it follows that as Note that

(2.7)

where

(2.8)

Thus, for by applying Lemma 2.2, we have

(2.9)

As for coupled with Markov's inequality and Lemma 2.1 with , then an application of Lemma 2.2 provides

(2.10)

Hence (2.6) holds true.

Proposition 2.5.

For and , one has uniformly

(2.11)

Proof.

Note that for large enough,

(2.12)

when , uniformly for

Proposition 2.6.

Under the assumptions of Theorem 1.1, one has

(2.13)

Proof.

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

(2.14)

Proof of Theorem 1.1.

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

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

(3.1)

Proposition 3.2.

For and one has

(3.2)

where

Proposition 3.3.

For and one has

(3.3)

Proposition 3.4.

For and one has

(3.4)