The Renyi Capacity and Center

TL;DR

This paper investigates the properties of Renyi's information measures, proves a conjecture for all positive orders, and explores the continuity and applications of Renyi capacity and center in probability measures and Poisson processes.

Contribution

It proves the van Erven-Harremoes conjecture for all positive orders and generalizes it, establishing new continuity and uniform equicontinuity results for Renyi capacity and information.

Findings

Proved the van Erven-Harremoes conjecture for all positive orders.

Established continuity and uniform equicontinuity of Renyi capacity.

Derived Renyi capacities and centers for various Poisson process families.

Abstract

Renyi's information measures ---the Renyi information, mean, capacity, radius, and center--- are analyzed relying on the elementary properties of the Renyi divergence and the power means. The van Erven-Harremoes conjecture is proved for any positive order and for any set of probability measures on a given measurable space and a generalization of it is established for the constrained variant of the problem. The finiteness of the order $\alpha$ Renyi capacity is shown to imply the continuity of the Renyi capacity on $(0,\alpha]$ and the uniform equicontinuity of the Renyi information, both as a family of functions of the order indexed by the priors and as a family of functions of the prior indexed by the orders. The Renyi capacities and centers of various families of Poisson processes are derived as examples.