Modern Stochastics: Theory and Applications

In this article, we first obtain, for the Kolmogorov distance, an error bound between a tempered stable and a compound Poisson distribution (CPD) and also an error bound between a tempered stable and an α-stable distribution via Stein’s method. For the smooth Wasserstein distance, an error bound between two tempered stable distributions (TSDs) is also derived. As examples, we discuss the approximation of a TSD to normal and variance-gamma distributions (VGDs). As corollaries, the corresponding limit theorem follows.

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.