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

Benjamin Arras, Christian Houdré (Autoren)

Buch | Softcover

XI, 104 Seiten

2019 | 1st ed. 2019
Springer International Publishing (Verlag)
978-3-030-15016-7 (ISBN)

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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.

1 Introduction.- 2 Preliminaries.- 3 Characterization and Coupling.- 4 General Upper Bounds by Fourier Methods.- 5 Solution to Stein's Equation for Self-Decomposable Laws.- 6 Applications to Sums of Independent Random Variables.

"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)

Erscheinungsdatum	28.04.2019
Reihe/Serie	SpringerBriefs in Probability and Mathematical Statistics
Zusatzinfo	XI, 104 p. 1 illus.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	189 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
Schlagworte	Infinite Divisibility • Kolmogorov Distance • Rates of Convergence • Self-decomposability • Smooth Wasserstein Distance • Stable Laws • Stein's method • Stein-Thikhomirov's Method • Weak Limit Theorems
ISBN-10	3-030-15016-X / 303015016X
ISBN-13	978-3-030-15016-7 / 9783030150167
Zustand	Neuware