Coding on countably infinite alphabets
Stephane Boucheron, Aurelien Garivier and Elisabeth Gassiat
IEEE Transactions on Information Theory
This paper describes universal lossless coding strategies for compressing sources on countably infinite alphabets.
Classes of memoryless sources defined by an envelope condition on the marginal distribution provide benchmarks
for coding techniques originating from the theory of universal coding over finite alphabets. We prove general upperbounds
on minimax regret and lower-bounds on minimax redundancy for such source classes. The general upper
bounds emphasize the role of the Normalized Maximum Likelihood codes with respect to minimax regret in the
infinite alphabet context. Lower bounds are derived by tailoring sharp bounds on the redundancy of Krichevsky-
Trofimov coders for sources over finite alphabets. Up to logarithmic (resp. constant) factors the bounds are matching
for source classes defined by algebraically declining (resp. exponentially vanishing) envelopes. Effective and (almost)
adaptive coding techniques are described for the collection of source classes defined by algebraically vanishing
envelopes. Those results extend our knowledge concerning universal coding to contexts where the key tools from
parametric inference (Bernstein-Von Mises theorem, Wilks theorem) are known to fail.