On Calculation of Constants in the Arak Inequalities for Concentration Functions of Convolutions of Probability Distributions

Abstract

The aim of the present work is evaluation of absolute constants in the Arak inequalities for the concentration functions of convolutions of probability distributions. This result allows us to calculate the constant in the inequality for the uniform distance between n and (n + 1)-fold convolutions of one-dimensional symmetric probability distributions with a characteristic function separated from −1, as well as a number of other estimates, in particular, the accuracy of approximation of samples of rare events by the Poisson point process.