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

  • Benjamin Arras
  • Christian Houdré

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

Introduction

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.

Keywords

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

  • Benjamin Arras
    • 1
  • Christian Houdré
    • 2
  1. 1.Laboratoire Paul PainlevéUniversity of Lille Nord de FranceVilleneuve-d’AscqFrance
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-030-15017-4
  • Copyright Information The Author(s), under exclusive license to Springer Nature Switzerland AG 2019
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-15016-7
  • Online ISBN 978-3-030-15017-4
  • Series Print ISSN 2365-4333
  • Series Online ISSN 2365-4341
  • Buy this book on publisher's site