Berry-Esseen bounds for functionals of independent random variables

TL;DR

This paper establishes Berry-Esseen bounds for a broad class of functionals of independent random variables, including U-statistics and quadratic forms, improving existing bounds and extending applicability.

Contribution

It introduces new Kolmogorov distance bounds for these functionals using chaos expansion methods, covering cases with non-degenerate and degenerate structures.

Findings

Provides sharper bounds for quadratic forms, including matrices with diagonals.

Extends Berry-Esseen bounds to weighted U-statistics and Hoeffding decompositions.

Achieves bounds with fourth moment conditions, broadening previous results.

Abstract

We derive Berry-Esseen approximation bounds for general functionals of independent random variables, based on chaos expansions methods. Our results apply to $U$-statistics satisfying the weak assumption of decomposability in the Hoeffding sense, and yield Kolmogorov distance bounds instead of the Wasserstein bounds previously derived in the special case of degenerate $U$-statistics. Linear and quadratic functionals of arbitrary sequences of independent random variables are included as particular cases, with new fourth moment bounds, and applications are given to Hoeffding decompositions, weighted $U$-statistics, quadratic forms, and random subgraph weighing. In the case of quadratic forms, our results recover and improve the bounds available in the literature, and apply to matrices with non-empty diagonals.