On closeness of two discrete weighted sums

Abstract

The effect that weighted summands have on each other in approximations of S = w1S1 + w2S2 + · · · + wN SN is investigated. Here, Si's are sums of integer-valued random variables, and wi denote weights, i = 1, . . . , 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 Si has the Markov Binomial distribution. The Kolmogorov metric is used to estimate the accuracy of approximation.