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Fixed-rate lossless coding for AMS ergodic sources

Statement

Assume AC. For a finite-alphabet AMS source with ergodic stationary mean, fixed-rate lossless coding has vanishing block error above the stationary-mean entropy rate and success probability tending to zero below it.

Facts & Assumptions

Given: AC; a finite-alphabet AMS source with ergodic stationary mean and entropy rate h; and a target rate R≥0 with the fixed-length code convention of Fixed-length lossless block codes and error probability.

[A1]

The Axiom of Choice: AC is inherited only through the AMS information-AEP tail-domination prerequisite.

[F1]

Shannon–McMillan limit for finite-alphabet AMS ergodic sources: normalized self-information under the AMS block law converges almost surely, hence in probability, to h.

[F2]

AEP criterion for fixed-rate lossless coding: an information AEP gives the fixed-rate achievability and converse.

[F3]

Dominated convergence: bounded almost-surely convergent indicators converge in expectation.

Proof

Given: AC and the source and hypotheses above.

1.1A1F1F3givenalgebra

Almost-sure convergence in [F1] implies convergence in probability by [F3] applied to the bounded deviation indicators.

2.1F1F2step 1.1∎

Apply [F2]. Every rate R>h admits fixed-length codes with block error tending to zero, and for every R<h every fixed-length code sequence has success probability tending to zero. This conclusion concerns the AMS source law; no stationarity assumption on that law is added.

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