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

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.