The Structure of the Realizations of the Causal Information Rate-Distortion Function for Markovian Sources: Realizations with Densities

TL;DR

This paper proves that for Markovian sources with density functions, the minimal memory length in causal IRDF realizations equals the source order, linking the structure of realizations to source properties.

Contribution

It establishes that the minimal integer in the causal IRDF realization equals the Markov order for sources with densities, clarifying the structure of such realizations.

Findings

The minimal integer f for the realization equals the source order f.

The result applies to sources with joint probability density functions.

The structure of realizations is characterized by specific Markov properties.

Abstract

The main purpose of this note is to show that in a realization $(x_{1}^{n}, y_{1}^{n})$ of the causal \textit{information rate-distortion function} (IRDF) for a $\kappa$-th order Markovian source $x_{1}^{n}$, under a single letter sum distortion constraint, the smallest integer $\ell$ for which $y_{k} \leftrightarrow y_{1}^{k-1},x_{k-\ell+1}^{k} \leftrightarrow x_{1}^{k-\ell}$ holds is $\ell=\kappa$. This result is derived under the assumption that the sequences $(x_{1}^{n},y_{1}^{n})$ have a joint probability density function.