On closeness of two discrete weighted sums

The effect that weighted summands have on each other in approximations of $S=w_1{S}_1+w_2{S}_2+\cdots +w_N{S}_N$ 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.