On Universal D-Semifaithful Coding for Memoryless Sources With Infinite Alphabets

The problem of variable length and fixed-distortion universal source coding (or D-semifaithful source coding) for stationary and memoryless sources on countably infinite alphabets (
$\infty $
-alphabets) is addressed in this paper. The main results of this work offer a set of sufficient conditions (from weaker to stronger) to obtain weak minimax universality, strong minimax universality, and corresponding achievable rates of convergences for the worst-case redundancy for the family of stationary memoryless sources whose densities are dominated by an envelope function (or the envelope family) on
$\infty $
-alphabets. An important implication of these results is that universal D-semifaithful source coding is not feasible for the complete family of stationary and memoryless sources on
$\infty $
-alphabets. To demonstrate this infeasibility, a sufficient condition for the impossibility is presented for the envelope family. Interestingly, it matches the well-known impossibility condition in the context of lossless (variable-length) universal source coding. More generally, this work offers a simple description of what is needed to achieve universal D-semifaithful coding for a family of distributions
$\Lambda $
. This reduces to finding a collection of quantizations of the product space at different block-lengths — reflecting the fixed distortion restriction — that satisfy two asymptotic requirements: the first is a universal quantization condition with respect to
$\Lambda $
, and the second is a vanishing information radius (I-radius) condition for
$\Lambda $
reminiscent of the condition known for lossless universal source coding.