Stein's Method for Tempered Stable Distributions

Abstract

In this article, we develop Stein characterization for two-sided tempered stable distributions using its characteristic function. It enables us to give the Stein characterizations for normal, gamma, Laplace, product of two normal, difference of two gamma, and variance-gamma distributions from the existing literature. Further, it also enables us to give the Stein characterizations for truncated Lévy flight, CGMY, KoBol, and bilateral-gamma distributions. We prove the existence of additive size bias for the one-sided case of tempered stable distributions, in particular, the gamma distribution. We also show that the Stein characterization for the convolution of independent tempered stable distributions can be derived from its characteristic function. Finally, we derive an error bound for tempered stable approximation.