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.

Reducibility of a finite Markov chain

Definition

Let $E$ be a finite nonempty state space and $p$ a transition matrix on $E$ . A subset $C \subseteq E$ is closed when $\sum_{y \in C} p (x, y) = 1$ for every $x \in C$ . The finite chain is reducible when it has a nonempty proper closed subset. This definition is the only reducibility convention used by the finite-support example on this page.

Depends on

Transition matrices and n-step probabilities

Used by

Dependency tree · two levels

3 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

Art B. Owen, Monte Carlo Theory, Methods and Examples, Chapter 12 (standard reference, not scraped)
Gareth O. Roberts and Jeffrey S. Rosenthal, General State Space Markov Chains and MCMC Algorithms (standard reference, not scraped)