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

1. Front Matter

   Pages i-xi

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2. Introduction

   • Benjamin Arras, Christian Houdré

   Pages 1-2

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3. Preliminaries

   • Benjamin Arras, Christian Houdré

   Pages 3-12

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4. Characterization and Coupling

   • Benjamin Arras, Christian Houdré

   Pages 13-29

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5. General Upper Bounds by Fourier Methods

   • Benjamin Arras, Christian Houdré

   Pages 31-56

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6. Solution to Stein’s Equation for Self-Decomposable Laws

   • Benjamin Arras, Christian Houdré

   Pages 57-75

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7. Applications to Sums of Independent Random Variables

   • Benjamin Arras, Christian Houdré

   Pages 77-88

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8. Back Matter

   Pages 89-104

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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 classicalweak 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

“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)

• Laboratoire Paul Painlevé, University of Lille Nord de France, Villeneuve-d’Ascq, France

  Benjamin Arras

• School of Mathematics, Georgia Institute of Technology, Atlanta, USA

  Christian Houdré

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