Approximations Related to Tempered Stable Distributions

TL;DR

This paper derives error bounds for approximating tempered stable distributions with various other distributions using Stein's method, providing theoretical insights and practical examples.

Contribution

It introduces new error bounds for approximations of tempered stable distributions to other distributions, including stable, Poisson, normal, and variance gamma, using Stein's method.

Findings

Error bounds for Kolmogorov distance between tempered stable and compound Poisson distributions.

Error bounds for Kolmogorov distance between tempered stable and alpha stable distributions.

Error bounds for smooth Wasserstein distance between two tempered stable distributions.

Abstract

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