Concentration bounds for entropy estimation of one-dimensional Gibbs measures

TL;DR

This paper derives concentration bounds for two entropy estimators of one-dimensional Gibbs measures, providing probabilistic guarantees on their fluctuations based on a new exponential inequality.

Contribution

It introduces novel concentration bounds for entropy estimators of Gibbs measures using a general exponential inequality for Lipschitz functions.

Findings

Bounds on fluctuations of entropy estimators are established.

The results apply to estimators based on empirical frequencies and block appearances.

The bounds are exponential inequalities demonstrating the estimators' stability.

Abstract

We obtain bounds on fluctuations of two entropy estimators for a class of one-dimensional Gibbs measures on the full shift. They are the consequence of a general exponential inequality for Lipschitz functions of n variables. The first estimator is based on empirical frequencies of blocks scaling logarithmically with the sample length. The second one is based on the first appearance of blocks within typical samples.