A convolution inequality, yielding a sharper Berry-Esseen theorem for summands Zolotarev-close to normal

TL;DR

This paper improves the classical Berry-Esseen bound by using a weak norm distance to normality, leading to sharper and more straightforward error estimates for sums of i.i.d. random variables.

Contribution

It introduces a new convolution inequality that refines the Berry-Esseen theorem, combining Zolotarev's approximation with a novel bound on the Kolmogorov distance between convolutions.

Findings

Sharper Berry-Esseen bounds using weak norm distances

A new convolution inequality bounding Kolmogorov distance

Simplified inequalities for probability distances

Abstract

The classical Berry-Esseen error bound, for the normal approximation to the law of a sum of independent and identically distributed random variables, is here improved by replacing the standardised third absolute moment by a weak norm distance to normality. We thus sharpen and simplify two results of Ulyanov (1976) and of Senatov (1998), each of them previously optimal, in the line of research initiated by Zolotarev (1965) and Paulauskas (1969). Our proof is based on a seemingly incomparable normal approximation theorem of Zolotarev (1986), combined with our main technical result: The Kolmogorov distance (supremum norm of difference of distribution functions) between a convolution of two laws and a convolution of two Lipschitz laws is bounded homogeneously of degree 1 in the pair of the Wasserstein distances (L$^1$ norms of differences of distribution functions) of the corresponding factors, and also, inessentially for the present application, in the pair of the Lipschitz constants. Side results include a short introduction to $\zeta$ norms on the real line, simpler inequalities for various probability distances, slight improvements of the theorem of Zolotarev (1986) and of a lower bound theorem of Bobkov, Chistyakov and G\"otze (2012), an application to sampling from finite populations, auxiliary results on rounding and on winsorisation, and computations of a few examples. The introductory section in particular is aimed at analysts in general rather than specialists in probability approximations.