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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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"This monograph is an excellent starting point for researchers to explore this fascinating area." (Fraser Daly, zbMATH 1447.60052, 2020)
"The book is interesting and well written. It may be recommended as a must-have item to the researchers interested in limit theorems of probability theory as well as to other probability theorists." (Przemyslaw matula, Mathematical Reviews, January, 2020)
"The book is interesting and well written. It may be recommended as a must-have item to the researchers interested in limit theorems of probability theory as well as to other probability theorists." (Przemyslaw matula, Mathematical Reviews, January, 2020)