© 2019

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


Table of contents

  1. Front Matter
    Pages i-xi
  2. Benjamin Arras, Christian Houdré
    Pages 1-2
  3. Benjamin Arras, Christian Houdré
    Pages 3-12
  4. Benjamin Arras, Christian Houdré
    Pages 13-29
  5. Benjamin Arras, Christian Houdré
    Pages 31-56
  6. Benjamin Arras, Christian Houdré
    Pages 57-75
  7. Benjamin Arras, Christian Houdré
    Pages 77-88
  8. Back Matter
    Pages 89-104

About this book


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.


Infinite Divisibility Self-decomposability Stable Laws Stein's method Stein-Thikhomirov's Method Weak Limit Theorems Rates of Convergence Kolmogorov Distance Smooth Wasserstein Distance

Authors and affiliations

  1. 1.Laboratoire Paul PainlevéUniversity of Lille Nord de FranceVilleneuve-d’AscqFrance
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

Bibliographic information

  • Book Title On Stein's Method for Infinitely Divisible Laws with Finite First Moment
  • Authors Benjamin Arras
    Christian Houdré
  • Series Title SpringerBriefs in Probability and Mathematical Statistics
  • Series Abbreviated Title SpringerBriefs in Probabil., Math.Statist.
  • DOI
  • Copyright Information The Author(s), under exclusive license to Springer Nature Switzerland AG 2019
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-030-15016-7
  • eBook ISBN 978-3-030-15017-4
  • Series ISSN 2365-4333
  • Series E-ISSN 2365-4341
  • Edition Number 1
  • Number of Pages XI, 104
  • Number of Illustrations 1 b/w illustrations, 0 illustrations in colour
  • Topics Probability Theory and Stochastic Processes
  • Buy this book on publisher's site
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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.” (Przemysław matuła, Mathematical Reviews, January, 2020)