Asymptotic Expansions for Symmetric Statistics with Degenerate Kernels

Abstract

Asymptotic expansions for U-statistics and V-statistics with degenerate kernels are investigated, respectively, and the remainder term O(n1−p/2), for some p≥4, is shown in both cases. From the results, it is obtained that asymptotic expansions for the Crame´r–von Mises statistics of the uniform distribution U(0,1) hold with the remainder term On1−p/2 for any p≥4. The scheme of the proof is based on three steps. The first one is the almost sure convergence in a Fourier series expansion of the kernel function u(x,y). The key condition for the convergence is the nuclearity of a linear operator Tu defined by the kernel function. The second one is a representation of U-statistics or V-statistics by single sums of Hilbert space valued random variables. The third one is to apply asymptotic expansions for single sums of Hilbert space valued random variables.