Empirical processes, typical sequences and coordinated actions in standard Borel spaces

TL;DR

This paper introduces a new notion of typical sequences for standard Borel spaces based on empirical distribution approximations, enabling derivation of fundamental limits in source coding for complex alphabets.

Contribution

It extends the concept of typicality to general Borel spaces using Glivenko-Cantelli classes, facilitating analysis of source coding with abstract alphabets.

Findings

Unified framework for typical sequences in Borel spaces

Simplified derivations of source coding limits

Applicable to a wide class of measurable functions

Abstract

This paper proposes a new notion of typical sequences on a wide class of abstract alphabets (so-called standard Borel spaces), which is based on approximations of memoryless sources by empirical distributions uniformly over a class of measurable "test functions." In the finite-alphabet case, we can take all uniformly bounded functions and recover the usual notion of strong typicality (or typicality under the total variation distance). For a general alphabet, however, this function class turns out to be too large, and must be restricted. With this in mind, we define typicality with respect to any Glivenko-Cantelli function class (i.e., a function class that admits a Uniform Law of Large Numbers) and demonstrate its power by giving simple derivations of the fundamental limits on the achievable rates in several source coding scenarios, in which the relevant operational criteria pertain to reproducing empirical averages of a general-alphabet stationary memoryless source with respect to a suitable function class.