This book describes extensions of Sudakov's classical result on the concentration of measure phenomenon for weighted sums of dependent random variables. The central topics of the book are weighted sums of random variables and the concentration of their distributions around Gaussian laws. The analysis takes place within the broader context of concentration of measure for functions on high-dimensional spheres. Starting from the usual concentration of Lipschitz functions around their limiting mean, the authors proceed to derive concentration around limiting affine or polynomial functions, aiming towards a theory of higher order concentration based on functional inequalities of log-Sobolev and Poincare type. These results make it possible to derive concentration of higher order for weighted sums of classes of dependent variables.

While the first part of the book discusses the basic notions and results from probability and analysis which are needed for the remainder of the book, the latter parts provide a thorough exposition of concentration, analysis on the sphere, higher order normal approximation and classes of weighted sums of dependent random variables with and without symmetries.

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Reviews

"The exposition in the book is careful and thorough, with proofs given throughout. Each chapter concludes with remarks to give context, references and recent developments in the topic at hand, making this an excellent starting point for a study of the randomised sums considered here. While they begin with fundamental results in the area, the authors move carefully and efficiently to state-of-the-art techniques and results. Overall, this is a well-written book that will prove invaluable to researchers in the area." (Fraser Daly, zbMATH 1542.60001, 2024)

"The book under review is a much-anticipated item by researchers, especially because it focuses on dependent random variables. It is worthy of recommendation to all those dealing with limit theorems for independent and dependent random variables and those studying the properties of multivariate random vectors." (Przemyslaw Matula, Mathematical Reviews, November, 2024)

Book Information
ISBN 9783031311482
Author Sergey Bobkov
Format Hardback
Page Count 434
Imprint Springer International Publishing AG
Publisher Springer International Publishing AG