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.
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