LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generated

How statement and proof provenance work

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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The DMC product laws are stochastic kernels

Statement

For a finite DMC $W$ and every $n \geq 1$ , the block formula $W^{n} (y^{n} ∣ x^{n}) = \prod_{i = 1}^{n} W (y_{i} ∣ x_{i})$ is a stochastic kernel from $X^{n}$ to $Y^{n}$ .

Facts & Assumptions

Given: A finite DMC $W$ with finite nonempty alphabets and an integer $n \geq 1$ .

[F1]

Finite discrete memoryless channels: the block values are $W^{n} (y^{n} ∣ x^{n}) = \prod_{i = 1}^{n} W (y_{i} ∣ x_{i})$ .

[F2]

Finite stochastic kernels: every entry of $W$ is nonnegative and $\sum_{y \in Y} W (y ∣ x) = 1$ for each input $x$ .

Proof

1.1F1F2given

Fix $x^{n} \in X^{n}$ . By [F1] and [F2], each product $W^{n} (y^{n} ∣ x^{n})$ is nonnegative.

2.1F1F2step 1.1algebra∎

Summing over the finite product alphabet and factoring the finite sum gives $\sum_{y^{n} \in Y^{n}} W^{n} (y^{n} ∣ x^{n}) = \prod_{i = 1}^{n} (\sum_{y_{i} \in Y} W (y_{i} ∣ x_{i})) = \prod_{i = 1}^{n} 1 = 1$ . Together with step 1.1, these are exactly the stochastic-kernel conditions.

Depends on

Used by

Cited to discharge well-definedness by Finite discrete memoryless channels.

Dependency tree · two levels

4 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 and Yihong Wu, Information Theory: From Coding to Learning, Chapter 19 §19.1 (standard reference, not scraped)
Robert Gallager, Principles of Digital Communication I, Chapter 8 §8.7.1, MIT OpenCourseWare (standard reference, not scraped)