Asymptotic Expansions for Symmetric Statistics with Degenerate Kernels

Asymptotic expansions for U-statistics and V-statistics with degenerate kernels are investigated, respectively, and the remainder term O(n(1-p/2)), for some p >= 4, is shown in both cases. From the results, it is obtained that asymptotic expansions for the Cramer-von Mises statistics of the uniform distribution U(0,1) hold with the remainder term O(n(1-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 T-u 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.

Automated summary

This study investigates asymptotic expansions for U-statistics and V-statistics with degenerate kernels, showing remainder terms of O(n(1-p/2)) in both cases. The results also suggest that asymptotic expansions for the Cramer-von Mises statistics of the uniform distribution U(0,1) are valid with the remainder term O(n(1-p/2) for any p >= 4. The proof scheme is based on three steps: almost sure convergence in a Fourier series expansion, representation of statistics by single sums of Hilbert space valued random variables, and application of asymptotic expansions for single sums of Hilbert space valued random variables.