Zero-Delay Lossy Coding of Linear Vector Markov Sources: Optimality of Stationary Codes and Near Optimality of Finite Memory Codes

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.