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 classical
weak limit theorems for sums of independent random variables. This book is aimed at graduate
students and researchers working in probability theory and mathematical statistics.
