Concentration of the information in data with log-concave distributions

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.