Almost sure limit theorems for U-statistics

Abstract

We relax the moment conditions from a result in almost sure limit theory for U-statistics due to Berkes and Csaki [(Stochastic Process. Appl. 94 (2001) 105)]. We extend this result to the case of convergence to stable laws and also prove a functional version.

Introduction

U-statistics generalize the concept of the sample mean of independent identically distributed (i.i.d.) random variables. The statistical interest in U-statistics stems from the fact that they form a class of unbiased estimators of a certain parameter with minimal variances. We begin by introducing some notation and recalling the concept of U-statistics.

Let X₁,X₂,… be i.i.d random variables with common distribution function F(x). Let m⩾1 and let $\text{h}\text{:}\text{R}{}^{\mathrm{m}}\text{→}\text{R}$ be a measurable function symmetric in its arguments. The U-statistic with kernel h is defined by

The kernel h is called degenerate with respect to F(x) if for all 1⩽j⩽m,

$$\text{∫}\text{R}\text{h(x}{}_1\text{,…,x}{}_{\mathrm{m}}\text{)}\text{d}\text{F(x}{}_{\mathrm{j}}\text{)≡0,}\text{where}\text{−∞<x}{}_1\text{,…,x}{}_{\mathrm{j-1}}\text{,x}{}_{\mathrm{j+1}}\text{,…,x}{}_{\mathrm{m}}\text{<∞.}$$

Let

$$\text{θ=Eh(X}{}_1\text{,…,X}{}_{\mathrm{m}}\text{)}$$

and for i=0,…,m let

$$\text{h}\text{̄}{}_{\mathrm{i}}\text{(x}{}_1\text{,…,x}{}_{\mathrm{i}}\text{)=Eh(x}{}_1\text{,…,x}{}_{\mathrm{i}}\text{,X}{}_{\mathrm{i+1}}\text{,…,X}{}_{\mathrm{m}}\text{),}$$$$\text{ζ}{}_{\mathrm{i}}\text{=}\text{Var}\text{(h}{}_{\mathrm{i}}\text{(X}{}_1\text{,…,X}{}_{\mathrm{i}}\text{)).}$$

Note that $\text{h}\text{̄}{}_0\text{=θ}$ and $\text{h}\text{̄}{}_{\mathrm{m}}\text{(x}{}_1\text{,…,x}{}_{\mathrm{m}}\text{)=h(x}{}_1\text{,…,x}{}_{\mathrm{m}}\text{)}$. Furthermore, the h_i are degenerate for i=1,…,m (see Denker, 1985). If ζ₁>0 the U-statistic U_n(h) is called non-degenerate and degenerate otherwise. The smallest integer c for which ζ_c>0 is called the critical parameter of the U-statistic U_n(h). Without loss of generality we will assume θ=0 throughout the rest of this article.

The theory of U-statistics started to develop intensively after Hoeffding's (1948) fundamental article. He showed asymptotic normality of non-degenerate U-statistics under the assumption

$$\text{Eh}{}^2\text{(X}{}_1\text{,…,X}{}_{\mathrm{m}}\text{)<∞,}$$

using the following representation of U-statistics:

$$\text{U}{}_{\mathrm{n}}\text{(h)=mU}{}_{\mathrm{n}}\text{(h}{}_1\text{)+}\text{∑}\text{k=2}\text{m}\text{m}\text{k}\text{U}{}_{\mathrm{n}}\text{(h}{}_{\mathrm{k}}\text{),}$$

where mU_n(h₁) is a sum of i.i.d. random variables and U_n(h₂),…,U_n(h_m) are U-statistics with degenerate kernels.

In the degenerate case n^c/2U_n(h) weakly converges to a multiple Wiener integral whenever h is square integrable (see e.g. Denker, 1985). Here c is the critical parameter of U_n(h).

In the late 1980s, Brosamler (1988) and Schatte (1988) independently proved a new type of limit theorem. This type of statement extends the classical central limit theorem in the i.i.d. case to a pathwise version and is therefore called an almost sure central limit theorem (ASCLT). In the 1990s, many studies were done to prove almost sure limit theorems (ASLT) in different situations, for example in the case of independent but not necessarily identically distributed random variables (see Berkes and Dehling, 1993). Excellent surveys on this topic may be found in Atlagh and Weber (2000) as well as in Berkes (1998). Recently Berkes and Csaki (2001) obtained a general result in almost sure limit theory. They used it to prove almost sure versions of several classical limit theorems. In particular they stated the following theorem for U-statistics.

Theorem 1.1

Let c be the critical parameter of the U-statistic U_n(h). Under assumption (1),

where G is the limit distribution of n^c/2U_n(h) and C_G denotes the set of continuity points of G.

In the present note we relax the moment condition in Theorem 1.1 and extend the statement in two directions. First we will obtain an ASLT with stable limiting distribution for a non-degenerate U-statistic. Furthermore we extend Theorem 1.1 to a functional version.