DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite stochastic kernels

Definition

Let $X, Y$ be finite nonempty sets. A stochastic kernel from $X$ to $Y$ is a matrix $K = (K (y ∣ x))_{x \in X, y \in Y}$ such that $K (y ∣ x) \geq 0$ and $\sum_{y \in Y} K (y ∣ x) = 1$ for every $x \in X$ . The convention is input first in the condition and output first in the value $K (y ∣ x)$ .

For a probability vector $P = (p_{x})_{x \in X}$ , the output vector $P K$ on $Y$ is $(P K)_{y} : = \sum_{x \in X} p_{x} K (y ∣ x) .$ It is a probability vector because its coordinates are nonnegative and $\sum_{y} (P K)_{y} = \sum_{x} p_{x} \sum_{y} K (y ∣ x) = \sum_{x} p_{x} = 1$ .

Depends on

Finite probability vectors

Used by

Finite degraded wiretap channel Definition
Finite discrete memoryless channels Definition
Finite discrete memoryless wiretap channel and code model Definition
Finite DMC information density Definition
Finite-source privacy–utility region Definition
Physically degraded two-receiver discrete memoryless broadcast channel Definition
Two-user discrete memoryless multiple-access channel Definition
Capacity of the binary erasure channel Example
Capacity of the binary symmetric channel Example
Capacity-achieving output laws bound every row divergence Lemma
IID DMC information density converges to mutual information Lemma
Strictly feasible DMC input laws achieve every lower mutual-information rate Lemma
The DMC product laws are stochastic kernels Lemma
The finite wiretap product law is stochastic Lemma
The n-letter mutual information of a DMC is at most nC Lemma
Uniform concentration around the capacity output divergence bound Lemma
Finite data processing for f-divergences Theorem
Finite data processing for Kullback–Leibler divergence Theorem
Finite data processing for Rényi divergence Theorem
Random coding achieves every rate below finite DMC capacity Theorem

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Sources

Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, Chapter 7, §§7.1–7.2 (standard reference, not scraped)
Tim van Erven and Peter Harremoës, Rényi Divergence and Kullback-Leibler Divergence, §II-B (standard reference, not scraped)