Concentration inequalities. A nonasymptotic theory of independence. (English) Zbl 1279.60005

Boucheron, Stéphane; Lugosi, Gábor; Massart, Pascal

Oxford: Oxford University Press (ISBN 978-0-19-953525-5/hbk). x, 481 p. (2013).

This new monograph deals with concentration inequalities for functions of independent random variables. This field has seen intensive research over the last decades due to its importance in many fields of applications such as statistics, learning theory, discrete mathematics, statistical mechanics, random matrix theory, information theory and high-dimensional geometry. The book gives a self-contained introduction which includes surveys of topics such as concentration of sums of independent random variables, variance bounds, the entropy method and the transportation method.
The clear exposition from basic material up to recent sophisticated results and lucid writing style make the text a pleasure to read. Beginners as well as experienced scientists will profit equally from it. It will certainly become one of the standard references in the field.

Reviewer: Hilmar Mai (Berlin)

MSC:

60-02	Research exposition (monographs, survey articles) pertaining to probability theory
60G50	Sums of independent random variables; random walks
39B62	Functional inequalities, including subadditivity, convexity, etc.

Keywords:

concentration inequality; information inequalities; Sobolev inequality; entropy method; empirical processes