Bounds of the accuracy of the normal approximation to the distributions of random sums under relaxed moment conditions

TL;DR

This paper improves bounds on the accuracy of normal approximation for sums of independent random variables under relaxed moment conditions, extending results to various random sum distributions and providing explicit constants.

Contribution

It introduces new bounds for normal approximation accuracy under minimal moment assumptions and extends these bounds to diverse random sum distributions.

Findings

Bounds are improved under relaxed moment conditions.

Results extend to Poisson-binomial, binomial, and Poisson sums.

Explicit constants are provided for approximation accuracy.

Abstract

Bounds of the accuracy of the normal approximation to the distribution of a sum of independent random variables are improved under relaxed moment conditions, in particular, under the absence of moments of orders higher than the second. These results are extended to Poisson-binomial, binomial and Poisson random sums. Under the same conditions, bounds are obtained for the accuracy of the approximation of the distributions of mixed Poisson random sums by the corresponding limit law. In particular, these bounds are constructed for the accuracy of approximation of the distributions of geometric, negative binomial and Poisson-inverse gamma (Sichel) random sums by the Laplace, variance gamma and Student distributions, respectively. All absolute constants are written out explicitly.