On the Existence of the Augustin Mean

TL;DR

This paper proves the existence and invariance of the Augustin mean for channels with countably generated output sigma-algebras, using convex analysis and operator theory.

Contribution

It establishes the existence of the Augustin mean for a broad class of channels and introduces a new family of operators to prove invariance for orders greater than one.

Findings

Existence of the Augustin mean is proven for channels with finite Augustin information.

A new family of operators is proposed to demonstrate invariance of the Augustin mean.

Some operators strictly decrease the conditional Re9nyi divergence unless at a fixed point.

Abstract

The existence of a unique Augustin mean and its invariance under the Augustin operator are established for arbitrary input distributions with finite Augustin information for channels with countably generated output $\sigma$-algebras. The existence is established by representing the conditional R\'enyi divergence as a lower semicontinuous and convex functional in an appropriately chosen uniformly convex space and then invoking the Banach--Saks property in conjunction with the lower semicontinuity and the convexity. A new family of operators is proposed to establish the invariance of the Augustin mean under the Augustin operator for orders greater than one. Some members of this new family strictly decrease the conditional R\'enyi divergence, when applied to the second argument of the divergence, unless the second argument is a fixed point of the Augustin operator.