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.
Categories: cs.IT math.IT