Convexity and robustness of the R\'{e}nyi entropy

TL;DR

This paper investigates the convexity and robustness properties of Rényi entropy on finite alphabets, analyzing its convergence behavior and relationships with Poisson and binomial distributions.

Contribution

It provides new insights into the convexity and robustness of Rényi entropy, including convergence results and distributional limits on finite alphabets.

Findings

Rényi entropy is convex as a function of alpha on finite alphabets.

The robustness of Rényi entropy depends on the initial alphabet size.

Convergence of disturbed entropy is established for increasing alphabet size.

Abstract

We study convexity properties of R\'{e}nyi entropy as function of $\alpha>0$ on finite alphabets. We also describe robustness of the R\'{e}nyi entropy on finite alphabets, and it turns out that the rate of respective convergence depends on initial alphabet. We establish convergence of the disturbed entropy when the initial distribution is uniform but the number of events increases to $\infty$ and prove that limit of R\'{e}nyi entropy of binomial distribution is equal to R\'{e}nyi entropy of Poisson distribution.