On the nonuniform Berry--Esseen bound

TL;DR

This paper improves bounds on the nonuniform Berry--Esseen inequality, introducing new Fourier-based methods and refinements to better estimate the constant factors involved in the tail zones.

Contribution

It proposes novel Fourier-based techniques and refinements to existing methods for deriving nonuniform Berry--Esseen bounds, narrowing the gap between known bounds.

Findings

Improved upper bounds on the nonuniform BE constant c_{nu}

Introduction of a Fourier-based method for tail zone analysis

Quick proof of Nagaev's nonuniform BE bound

Abstract

Due to the effort of a number of authors, the value c_u of the absolute constant factor in the uniform Berry--Esseen (BE) bound for sums of independent random variables has been gradually reduced to 0.4748 in the iid case and 0.5600 in the general case; both these values were recently obtained by Shevtsova. On the other hand, Esseen had shown that c_u cannot be less than 0.4097. Thus, the gap factor between the best known upper and lower bounds on (the least possible value of) c_u is now rather close to 1. The situation is quite different for the absolute constant factor c_{nu} in the corresponding nonuniform BE bound. Namely, the best correctly established upper bound on c_{nu} in the iid case is about 25 times the corresponding best known lower bound, and this gap factor is greater than 30 in the general case. In the present paper, improvements to the prevailing method (going back to S. Nagaev) of obtaining nonuniform BE bounds are suggested. Moreover, a new method is presented, of a rather purely Fourier kind, based on a family of smoothing inequalities, which work better in the tail zones. As an illustration, a quick proof of Nagaev's nonuniform BE bound is given. Some further refinements in the application of the method are shown as well.