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Fixed-rate source coding theorem for stationary ergodic sources

Statement

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

Facts & Assumptions

Given: A stationary ergodic finite-alphabet source of entropy rate h, a target rate R≥0, and fixed-length binary block codes as in Fixed-length lossless block codes and error probability.

[F1]

Ergodic Shannon–McMillan–Breiman theorem: −n−1log⁡2P(X0n−1)→h almost surely, hence in probability.

[F2]

AEP criterion for fixed-rate lossless coding: information convergence in probability gives vanishing error above h and vanishing success below h.

[F3]

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

Proof

Given: The source, its entropy rate and coding convention.

1.1F1F3givenalgebra

Almost-sure convergence in [F1] implies convergence in probability: for each ϵ>0, the indicators of deviations by more than ϵ tend almost surely to zero and are bounded by one, so [F3] makes their expectations tend to zero.

2.1F1F2step 1.1∎

Apply [F2] with this AEP limit. For every R>h there are rate-R codes with block error tending to zero; for every R<h, every rate-R code sequence has success probability tending to zero. At R=h no boundary conclusion follows from the theorem.

Depends on

Used by

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