The Berry-Esseen bound in de Jong's CLT

TL;DR

This paper establishes an optimal Berry-Esseen bound for normalized, degenerate U-statistics in de Jong's CLT, linking moment convergence and Lindeberg conditions to asymptotic normality.

Contribution

It provides the first optimal order Berry-Esseen bound for de Jong's CLT involving degenerate U-statistics, extending recent Wasserstein distance results.

Findings

Bound matches the optimal order of recent Wasserstein bounds.

Convergence of the fourth moment to three suffices for normality.

Lindeberg-Feller type condition is also sufficient.

Abstract

We prove a Berry-Esseen bound in de Jong's classical CLT for normalized, completely degenerate $U$-statistics, which says that the convergence of the fourth moment sequence to three and a Lindeberg-Feller type negligibility condition are sufficient for asymptotic normality. Our bound is of the same optimal order as the bound on the Wasserstein distance to normality that has recently been proved by D\"obler and Peccati (2017).