LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generated

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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DMC minimum and maximum dispersion are attained

Statement

For every finite DMC, the set $Π (W)$ of capacity-achieving input laws is nonempty and compact, and the function $P \mapsto V (P, W)$ is continuous on $Π (W)$ . Therefore $V_{\min} (W)$ and $V_{\max} (W)$ in Conditional information variance and DMC dispersion are attained.

Facts & Assumptions

Given: A finite DMC and the definitions of $C (W)$ , $Π (W)$ , and $V (P, W)$ .

[F1]

Finite DMC information capacity is attained: a capacity-achieving input law exists.

[F2]

The capacity-achieving output law is unique: all capacity-achieving inputs induce the same output law $Q^{*}$ .

[F3]

Capacity-achieving output laws bound every row divergence: every row is supported on $Q^{*}$ , and $D (W (\cdot ∣ x) ∥ Q^{*}) \leq C (W)$ .

[F4]

Conditional information variance and DMC dispersion: $V (P, W)$ is the $P$ -average of row conditional variances of $\ln [W (Y ∣ x) / (P W) (Y)]$ .

[F5]

For a nonempty subset of $R^{n}$ with $n \geq 1$ , compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent: the finite-dimensional probability simplex is compact, and a continuous function on a nonempty compact set attains its extrema.

[F6]

Finite mutual information and Finite distributions, entropy, divergence, and stochastic matrices: finite mutual information has the finite entropy expansion, with $0 \ln 0 = 0$ .

[F7]

The logarithm grows more slowly than every positive real power: $t \ln t \to 0$ as $t ↓ 0$ .

[F8]

Single-letter information capacity of a finite DMC: $C (W) = \max_{P} I (P, W)$ and $Π (W)$ is its maximizing level set.

Proof

1.1F5F6F7F8given

The input simplex is compact by [F5]. Expanding the KL sum in [F6] gives $I (P, W) = H (P W) - \sum_{x} P (x) H (W_{x})$ . This finite entropy formula is continuous: [F7] gives $t \ln t \to 0$ at zero. By [F8], $Π (W) = {P : I (P, W) = C (W)}$ is the maximizing level set and is closed in the simplex.

1.2F2F3given

For $P \in Π (W)$ , [F2] gives $P W = Q^{*}$ , and [F3] says each row is supported on $Q^{*}$ . On every reachable output $y$ , $Q^{*} (y) > 0$ : some row has $W (y ∣ x) > 0$ , and [F3] requires that row's support to lie in $Q^{*}$ . Consequently each row variance $v_{x} : = {Var}_{W (\cdot ∣ x)} \ln [W (Y ∣ x) / Q^{*} (Y)]$ is a finite constant independent of $P \in Π (W)$ .

2.1F1step 1.1

By [F1], $Π (W)$ is nonempty; step 1.1 makes it a closed subset of a compact set, hence compact.

3.1F4F5step 2.1step 1.2algebra∎

On $Π (W)$ , [F4] therefore becomes $V (P, W) = \sum_{x} P (x) v_{x}$ , a finite linear function of $P$ , hence continuous. Applying [F5] to the compact nonempty set from step 2.1 proves attainment of its minimum and maximum.

Depends on

Used by

Nondegenerate normal approximation for finite unconstrained DMCs Theorem

Cited to discharge well-definedness by Conditional information variance and DMC dispersion.

Dependency tree · two levels

34 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Yury Polyanskiy, H. Vincent Poor, and Sergio Verdú, Channel Coding Rate in the Finite Blocklength Regime (standard reference, not scraped)
Masahito Hayashi, Information Spectrum Approach to Second-Order Coding Rate in Channel Coding (standard reference, not scraped)