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.

Harris recurrence relative to an irreducibility measure

Definition

Let $P$ be $ϕ$ -irreducible with nonzero $σ$ -finite irreducibility measure $ϕ$ . It is Harris recurrent relative to $ϕ$ if for every $A \in X$ with $ϕ (A) > 0$ and every starting state $x$ , the first hitting time $τ_{A} = \inf {n \geq 0 : X_{n} \in A}$ satisfies $\Pr_{x} (τ_{A} < \infty) = 1$ . This is an all-start almost-sure hitting property; it is stronger than $ϕ$ -irreducibility, which requires only positive probability at some finite time.

Depends on

Phi-irreducibility for a general-state Markov kernel

Used by

Dependency tree · two levels

5 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

Gareth O. Roberts and Jeffrey S. Rosenthal, General State Space Markov Chains and MCMC Algorithms (standard reference, not scraped)