The rates of convergence for generalized entropy of the normalized sums of IID random variables

TL;DR

This paper establishes the convergence rates of generalized entropy measures, specifically Rényi and Tsallis entropies, for normalized sums of IID continuous variables towards their Gaussian limits.

Contribution

It provides the first sharp rates of convergence for generalized entropies of normalized sums of IID variables with bounded moments.

Findings

Rényi and Tsallis entropies converge to Gaussian limits

Sharp convergence rates are derived

Results apply to variables with bounded moments

Abstract

We consider the generalized differential entropy of normalized sums of independent and identically distributed (IID) continuous random variables. We prove that the R\'{e}nyi entropy and Tsallis entropy of order $\alpha\ (\alpha>0)$ of the normalized sum of IID continuous random variables with bounded moments are convergent to the corresponding R\'{e}nyi entropy and Tsallis entropy of the Gaussian limit, and obtain sharp rates of convergence.