Concentration of the information in data with log-concave distributions

Abstract

A concentration property of the functional ${-}\log f(X)$ is demonstrated, when a random vector X has a log-concave density f on $\mathbb{R}^n$. This concentration property implies in particular an extension of the Shannon-McMillan-Breiman strong ergodic theorem to the class of discrete-time stochastic processes with log-concave marginals.Comment: Published in at http://dx.doi.org/10.1214/10-AOP592 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org