On weak approximation of U-statistics

TL;DR

This paper studies the weak convergence of U-statistics, relaxing classical moment conditions and assuming the kernel's conditional expectation is in the domain of attraction of the normal law, broadening the scope of weak approximation results.

Contribution

It introduces weaker moment and regularity conditions for the weak convergence of U-statistics, extending classical results.

Findings

Weak convergence holds under 4/3 moments with a logarithmic correction.

Conditional expectation of the kernel in the domain of attraction of the normal law suffices.

Classical second-moment conditions are significantly relaxed.

Abstract

This paper investigates weak convergence of U-statistics via approximation in probability. The classical condition that the second moment of the kernel of the underlying U-statistic exists is relaxed to having 4/3 moments only (modulo a logarithmic term). 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).