Concentration of the information in data with log-concave distributions

TL;DR

This paper demonstrates a concentration property of the negative log-density for log-concave distributions, extending the Shannon-McMillan-Breiman theorem to processes with log-concave marginals, revealing new probabilistic insights.

Contribution

It introduces a concentration property for the negative log-density of log-concave distributions and extends a fundamental ergodic theorem to this class.

Findings

Establishes a concentration property for -log f(X) in log-concave distributions

Extends Shannon-McMillan-Breiman theorem to processes with log-concave marginals

Provides new probabilistic tools for analyzing log-concave data

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.