Self-Normalized Moderate Deviations for Degenerate U-Statistics.

Abstract

In this paper, we study self-normalized moderate deviations for degenerate U-statistics of order 2. Let {Xi,i≥1} be i.i.d. random variables and consider symmetric and degenerate kernel functions in the form h(x,y)=∑l=1∞λlgl(x)gl(y), where λl>0, Egl(X1)=0, and gl(X1) is in the domain of attraction of a normal law for all l≥1. Under the condition ∑l=1∞λl<∞ and some truncated conditions for {gl(X1):l≥1}, we show that logP(∑1≤i≠j≤nh(Xi,Xj)max1≤l<∞λlVn,l2≥xn2)∼-xn22 for xn→∞ and xn=o(n), where Vn,l2=∑i=1ngl2(Xi). As application, a law of the iterated logarithm is also obtained.