A Unified Approach to Stein's Method for Stable Distributions

TL;DR

This paper develops a unified Stein's method framework for stable distributions, providing new identities, error bounds, and convergence rates for $eta$-stable approximations, enhancing understanding of their probabilistic properties.

Contribution

It introduces a novel Stein identity for $eta$-stable distributions using Lévy process connections, and derives explicit error bounds and convergence rates.

Findings

Established Stein identities for $eta$-stable distributions.

Derived explicit error bounds for stable approximation.

Compared convergence rates with existing literature.

Abstract

In this article, we first review the connection between L\'evy processes and infinitely divisible random variables, and the classification of infinitely divisible distributions. Using this connection and the L\'evy-Khinchine representation of the characteristic function, we establish a Stein identity for an infinitely divisible random variable. The classification and slight modification in approach give us a Stein identity for an $\alpha$-stable random variable with $\alpha\in (0,2).$ Using fine regularity estimates for the solution to Stein equation, we derive error bounds for $\alpha$-stable approximations. We then apply these results to obtain rates of convergence. Finally, we compare these rates with the results available in the literature.