CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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An inactive DMC cost constraint recovers ordinary capacity

Statement

For a finite DMC with letter cost $c$ and budget $Γ \geq c_{\max}$ , the cost-constrained operational capacity equals the ordinary DMC capacity, with the same average-error convention and in nats.

Facts & Assumptions

Given: A finite DMC, a finite letter cost, and a budget at least the maximum letter cost.

[F1]

Finite DMC with a nonnegative input cost and feasible budget and Average-cost fixed-length codes and operational DMC capacity: each codeword's per-use average cost is an average of the letter costs and the information-capacity constraint is $E c (X) \leq Γ$ .

[F2]

Average-cost capacity formula for a finite DMC: the cost-constrained operational capacity equals $\max_{P_{X} : E c (X) \leq Γ} I (P_{X}; W)$ .

[F3]

Operational capacity formula for a finite DMC: ordinary average-error DMC capacity equals $\max_{P_{X}} I (P_{X}; W)$ .

Proof

1.1F1givenalgebra

Every letter has $c (x) \leq c_{\max} \leq Γ$ by [F1]. Therefore every ordinary code satisfies the block cost constraint, and every input distribution satisfies the one-letter cost constraint. Conversely a cost-feasible code or distribution is still an ordinary one, so the two code classes and feasible input sets coincide.

2.1F2F3step 1.1given∎

Apply [F2] and [F3] to these identical code and input classes. Their operational capacities agree; both use average error and natural-log nats by their definitions.

Depends on

Used by

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Sources

Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, Chapter 20 §20.1 (standard reference, not scraped)
Yury Polyanskiy, MIT 6.441 Information Theory, Chapter 17 §§17.1–17.2 (standard reference, not scraped)