Bounds for the concentration functions of random sums under relaxed moment conditions

TL;DR

This paper derives bounds for the deviation of concentration functions of sums of independent random variables from the folded normal distribution, extending results to various random sum models without requiring higher-order moments.

Contribution

It provides new bounds for concentration functions under relaxed moment conditions and extends these results to multiple types of random sums and their limit distributions.

Findings

Bounds for deviation from folded normal distribution derived

Extensions to Poisson-binomial, binomial, and Poisson sums achieved

Explicit numerical constants provided for all bounds

Abstract

Estimates are constructed for the deviation of the concentration functions of sums of independent random variables with finite variances from the folded normal distribution function without any assumptions concerning the existence of the moments of summands of higher orders. The obtained results are extended to Poisson-binomial, binomial and Poisson random sums. Under the same assumptions, the bounds are obtained for the approximation of the concentration functions of mixed Poisson random sums by the corresponding limit distributions. In particular, bounds are obtained for the accuracy of approximation of the concentration functions of geometric, negative binomial and Sichel random sums by the exponential, the folded variance gamma and the folded Student distribution. Numerical estimates of all the constants involved are written out explicitly.