Studentized Processes of U-statistics

TL;DR

This paper develops a uniform in probability approximation for Studentized processes of non-degenerate U-statistics, relaxing classical moment conditions and extending the applicability to kernels with only 5/3 moments and domain of attraction of the normal law.

Contribution

It introduces a new approximation result for Studentized U-statistics processes under weaker moment and distributional assumptions than classical methods.

Findings

Established uniform in probability approximation using Wiener process.

Relaxed moment condition from second to 5/3 moments.

Extended the domain of attraction condition for kernels.

Abstract

A uniform in probability approximation is established for Studentized processes of non degenerate U-statistics of order m greater or equal to 2 in terms of a standard Wiener process. The classical condition that the second moment of kernel of the underlying U-statistic exists is relaxed to having 5/3 moments. Furthermore, the conditional expectation of the kernel is only assumed to be in the domain of attraction of the normal law (instead of the classical two moment condition).