Information inequalities and a dependent Central Limit Theorem

TL;DR

This paper extends information-theoretic convergence results of the Central Limit Theorem to dependent variables under mixing conditions, using normal perturbations to control joint densities and strengthen existing entropy-based convergence results.

Contribution

It adapts information-theoretic methods to dependent variables with mixing conditions, providing stronger entropy convergence results than previous weak convergence findings.

Findings

Established entropy convergence for dependent variables under Rosenblatt mixing.

Introduced a normal perturbation technique to control joint densities.

Strengthened previous results by Takano, Carlen, and Soffer.

Abstract

We adapt arguments concerning information-theoretic convergence in the Central Limit Theorem to the case of dependent random variables under Rosenblatt mixing conditions. The key is to work with random variables perturbed by the addition of a normal random variable, giving us good control of the joint density and the mixing coefficient. We strengthen results of Takano and of Carlen and Soffer to provide entropy-theoretic, not weak convergence.