DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Iterated transition kernels

Definition

For a probability kernel $K$ on $(E, E)$ , let $I$ be the identity kernel $I (x, A) = 1_{A} (x)$ and define $K^{0} : = I, K^{n + 1} : = K^{n} K,$ where composition is in chronological order: $K^{n + 1} (x, A) = \int_{E} K (y, A) K^{n} (x, d y) .$ The identity map makes $I$ a probability kernel, including on a one-point space. Induction using the published kernel-composition lemma shows that every $K^{n}$ is a probability kernel. Associativity makes products of several copies of $K$ unambiguous. No conditional-expectation version is selected and no form of Choice is used.

Depends on

Used by

Chapman-Kolmogorov equations Theorem

Dependency tree · two levels

6 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

Levin, Peres, Wilmer, Markov Chains and Mixing Times (standard reference, not scraped)