On closeness of two discrete weighted sums

TL;DR

This paper investigates how weighted sums of integer-valued random variables approximate each other, focusing on independence and Markov Binomial cases, using the Kolmogorov metric to measure accuracy.

Contribution

It provides new bounds on the closeness of weighted sums under independence and Markov Binomial assumptions, using factorial moments and the Kolmogorov metric.

Findings

Closeness of weighted sums is characterized by factorial moment matching.

Explicit bounds are derived for the Kolmogorov distance in both cases.

Results improve understanding of approximation accuracy for sums of integer-valued variables.

Abstract

The effect that weighted summands have on each other in approximations of $S=w_1S_1+w_2S_2+\cdots+w_NS_N$ is investigated. Here, $S_i$'s are sums of integer-valued random variables, and $w_i$ denote weights, $i=1,\dots,N$. Two cases are considered: the general case of independent random variables when their closeness is ensured by the matching of factorial moments and the case when the $S_i$ has the Markov Binomial distribution. The Kolmogorov metric is used to estimate the accuracy of approximation.