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During the last two decades, concentration of measure has been a subject ofvarious exciting developments in convex geometry, functional analysis,statistical physics, high-dimensional statistics, probability theory,information theory, communications and coding theory, computer science, andlearning theory. One common theme which emerges in these fields isprobabilistic stability: complicated, nonlinear functions of a large number ofindependent or weakly dependent random variables often tend to concentratesharply around their expected values. Information theory plays a key role inthe derivation of concentration inequalities. Indeed, both the entropy methodand the approach based on transportation-cost inequalities are two majorinformation-theoretic paths toward proving concentration.  This brief survey is based on a recent monograph of the authors in theFoundations and Trends in Communications and Information Theory (onlineavailable at http://arxiv.org/pdf/1212.4663v8.pdf), and a tutorial given by theauthors at ISIT 2015. It introduces information theorists to three maintechniques for deriving concentration inequalities: the martingale method, theentropy method, and the transportation-cost inequalities. Some applications ininformation theory, communications, and coding theory are used to illustratethe main ideas.

Concentration of Measure Inequalities and Their Communication and Information-Theoretic Applications