Asymptotic Normality of $U$-Statistics is Equivalent to Convergence in the Wasserstein Distance

TL;DR

This paper demonstrates that the asymptotic normality of U-statistics is equivalent to their convergence in Wasserstein distance under common mild conditions, assuming data is strictly stationary and absolutely regular.

Contribution

It establishes an equivalence between asymptotic normality and Wasserstein convergence for U-statistics under standard assumptions.

Findings

Asymptotic normality is equivalent to Wasserstein convergence.

Results hold under mild conditions with stationary, absolutely regular data.

Provides theoretical foundation linking distributional convergence and normality.

Abstract

We prove the claim in the title under mild conditions which are usually satisfied when trying to establish asymptotic normality. We assume strictly stationary and absolutely regular data.