DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-sol)audited 2026-09-30

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.

Accessibility, communication, and irreducibility

Definition

For the countable transition matrix $p$ of Transition matrices and n-step probabilities, define accessibility by

$x \to y ⟺ p^{(n)} (x, y) > 0 for some n \in N_{0} .$

States communicate, written $x \leftrightarrow y$ , when $x \to y$ and $y \to x$ . The chain is irreducible when every pair of states communicates. Because $p^{(0)} (x, y) = 1_{{x = y}}$ , every state is accessible from itself with a zero-step path.

Depends on

Transition matrices and n-step probabilities

Used by

Irreducible recurrence/transience dichotomy Corollary
Different classes can have different recurrence types Counterexample
Aperiodic irreducible chain Definition
Birth–death recurrence through scale products Example
Communicating classes in a four-state chain Example
Laziness makes an irreducible chain aperiodic Example
Period two on a bipartite graph Example
Communication is an equivalence relation Lemma
Geometric tail for hitting in a finite irreducible chain Lemma
Period is constant on communicating classes Lemma
Recurrence and transience are class properties Theorem

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

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