  Optimal zero-delay coding (quantization) of $\mathbb{R}^d$-valued linearly
generated Markov sources is studied under quadratic distortion. The structure
and existence of deterministic and stationary coding policies that are optimal
for the infinite horizon average cost (distortion) problem is established.
Prior results studying the optimality of zero-delay codes for Markov sources
for infinite horizons either considered finite alphabet sources or, for the
$\mathbb{R}^d$-valued case, only showed the existence of deterministic and
non-stationary Markov coding policies or those which are randomized. In
addition to existence results, for finite blocklength (horizon) $T$ the
performance of an optimal coding policy is shown to approach the infinite time
horizon optimum at a rate $O(\frac{1}{T})$. This gives an explicit rate of
convergence that quantifies the near-optimality of finite window
(finite-memory) codes among all optimal zero-delay codes.
