On Stein's Method for Infinitely Divisible Laws with Finite First Moment, 1st ed. 2019
SpringerBriefs in Probability and Mathematical Statistics Series

This book focuses on quantitative approximation results for weak limit theorems when the target limiting law is infinitely divisible with finite first moment. Two methods are presented and developed to obtain such quantitative results. At the root of these methods stands a Stein characterizing identity discussed in the third chapter and obtained thanks to a covariance representation of infinitely divisible distributions. The first method is based on characteristic functions and Stein type identities when the involved sequence of random variables is itself infinitely divisible with finite first moment. In particular, based on this technique, quantitative versions of compound Poisson approximation of infinitely divisible distributions are presented. The second method is a general Stein's method approach for univariate selfdecomposable laws with finite first moment. Chapter 6 is concerned with applications and provides general upper bounds to quantify the rate of convergence in classicalweak limit theorems for sums of independent random variables. This book is aimed at graduate students and researchers working in probability theory and mathematical statistics.

1 Introduction.- 2 Preliminaries.- 3 Characterization and Coupling.- 4 General Upper Bounds by Fourier Methods.- 5 Solution to Stein's Equation for Self-Decomposable Laws.- 6 Applications to Sums of Independent Random Variables.

Covers connections between infinite divisibility and Stein's method First to propose a general and unifying Stein's methodology for infinitely divisible law with finite first moment Provides quantitative versions of classical weak limit theories for sum of independent random variables