Optimal Nonstationary Reproduction Distribution for Nonanticipative RDF on Abstract Alphabets

TL;DR

This paper defines and analyzes a nonanticipative Rate Distortion Function on abstract alphabets, establishing its properties, deriving a recursive closed-form solution, and connecting it to operational coding theorems and nonanticipatory entropy.

Contribution

It introduces a formal definition for nonanticipative RDF on abstract alphabets and derives a recursive closed-form expression for the optimal nonstationary reproduction distribution.

Findings

Proved existence of optimal reproduction distribution.

Established properties like compactness and lower semicontinuity.

Derived a recursive formula for the optimal distribution.

Abstract

In this paper we introduce a definition for nonanticipative Rate Distortion Function (RDF) on abstract alphabets, and we invoke weak convergence of probability measures to show various of its properties, such as, existence of the optimal reproduction conditional distribution, compactness of the fidelity set, lower semicontinuity of the RDF functional, etc. Further, we derive the closed form expression of the optimal nonstationary reproduction distribution. This expression is computed recursively backward in time. Throughout the paper we point out an operational meaning of the nonanticipative RDF by recalling the coding theorem derive in \cite{tatikonda2000}, and we state relations to Gorbunov-Pinsker's nonanticipatory $\epsilon-$entropy \cite{gorbunov-pinsker}.