On Stein's Method for Infinitely Divisible Laws With Finite First Moment

TL;DR

This paper develops a unified Stein methodology for infinitely divisible laws with finite first moments, enabling explicit convergence rates and bounds in limit theorems for sums of independent variables.

Contribution

It introduces a novel non-local Stein operator for these laws, extending size bias concepts, and applies Fourier and semigroup techniques to derive quantitative convergence bounds.

Findings

Explicit rates of convergence for compound Poisson approximations.

Connections between size bias, zero bias, and infinite divisibility.

Quantitative bounds in weak limit theorems for sums of independent variables.

Abstract

We present, in a unified way, a Stein methodology for infinitely divisible laws (without Gaussian component) having finite first moment. Based on a correlation representation, we obtain a characterizing non-local Stein operator which boils down to classical Stein operators in specific examples. Thanks to this characterizing operator, we introduce various extensions of size bias and zero bias distributions and prove that these notions are closely linked to infinite divisibility. Combined with standard Fourier techniques, these extensions also allow obtaining explicit rates of convergence for compound Poisson approximation in particular towards the symmetric $\alpha$-stable distribution. Finally, in the setting of non-degenerate self-decomposable laws, by semigroup techniques, we solve the Stein equation induced by the characterizing non-local Stein operator and obtain quantitative bounds in weak limit theorems for sums of independent random variables going back to the work of Khintchine and L\'evy.