  This paper derives novel results on the characterization of the the causal
information rate-distortion function (IRDF) $R_{c}^{it}(D)$ for
arbitrarily-distributed one-sided stationary $\kappa$-th order Markov source
x(1),x(2),.... It is first shown that Gorbunov and Pinsker's results on the
stationarity of the realizations to the causal IRDF (stated for two-sided
stationary sources) do not apply to the commonly used family of asymptotic
average single-letter (AASL) distortion criteria. Moreover, we show that, in
general, a reconstruction sequence cannot be both jointly stationary with a
one-sided stationary source sequence and causally related to it. This implies
that, in general, the causal IRDF for one-sided stationary sources cannot be
realized by a stationary distribution. However, we prove that for an
arbitrarily distributed one-sided stationary source and a large class of
distortion criteria (including AASL), the search for $R_{c}^{it}(D)$ can be
restricted to distributions which yield the output sequence y(1), y(2),...
jointly stationary with the source after $\kappa$ samples. Finally, we improve
the definition of the stationary causal IRDF $\overline{R}_{c}^{it}(D)$
previously introduced by Derpich and {\O}stergaard for two-sided Markovian
stationary sources and show that $\overline{R}_{c}^{it}(D)$ for a two-sided
source ...,x(-1),x(0),x(1),... equals $R_{c}^{it}(D)$ for the associated
one-sided source x(1), x(2),.... This implies that, for the Gaussian quadratic
case, the practical zero-delay encoder-decoder pairs proposed by Derpich and
{\O}stergaard for approaching $R_{c}^{it}(D)$ achieve an operational data rate
which exceeds $R_{c}^{it}(D)$ by less than $1+0.5 \log_2(2 \pi e /12) \simeq
1.254$ bits per sample.
