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.

Period of a state

Definition

For $x \in E$ , let

$R_{x} : = {n \in N : n \geq 1 and p^{(n)} (x, x) > 0} .$

If $R_{x} = \emptyset$ , set $d (x) : = 0$ . If $R_{x} \neq \emptyset$ , define $d (x)$ to be the greatest positive integer dividing every element of $R_{x}$ (the gcd convention of Common divisor, and the greatest common divisor $\gcd (a, b)$ , with the convention $\gcd (0, 0) : = 0$ ). This greatest divisor exists: fix one $t \in R_{x}$ ; every common positive divisor of $R_{x}$ is a divisor of $t$ , so the set of such divisors is a nonempty finite subset of the positive divisors of $t$ , and therefore has a greatest element. The set being maximized is intrinsic to $R_{x}$ , so the result is independent of the auxiliary $t$ . Only positive return times enter; $p^{(0)} (x, x) = 1$ does not force $d (x) = 1$ .

Depends on

Used by

Aperiodic irreducible chain Definition
Laziness makes an irreducible chain aperiodic Example
Period two on a bipartite graph Example
Period is constant on communicating classes Lemma

Dependency tree · two levels

10 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)