Optimal Nonstationary Reproduction Distribution for Nonanticipative RDF on Abstract Alphabets

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}.