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.

Recurrent and transient states

Definition

For a specified Markov chain law with transition matrix $p$ and deterministic initial state $x$ , write $P_{x}$ for that law. The state $x$ is recurrent when

$P_{x} (T_{x}^{+} < \infty) = 1,$

and transient when

$P_{x} (T_{x}^{+} < \infty) < 1.$

Here $T_{x}^{+}$ is the strictly positive return time of Hitting, return, and visit times, so the initial visit at time zero does not count as a return. Since the displayed return probability lies in $[0, 1]$ , these alternatives exhaust all states. The definition concerns the specified law $P_{x}$ and does not assert that laws for every state can be selected simultaneously.

Depends on

Used by

Higher-dimensional simple symmetric walks are transient Corollary
Irreducible recurrence/transience dichotomy Corollary
One-dimensional simple symmetric walk is recurrent Corollary
Two-dimensional simple symmetric walk is recurrent Corollary
A transient chain can return with positive probability Counterexample
Different classes can have different recurrence types Counterexample
Birth–death recurrence through scale products Example
Communicating classes in a four-state chain Example
Green kernel of a biased integer walk Example
Geometric tail for hitting in a finite irreducible chain Lemma
Equivalent criteria for recurrence and transience Theorem
Recurrence and transience are class properties Theorem
Renewal decomposition at successive returns 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

Durrett, Probability: Theory and Examples, fifth edition (standard reference, not scraped)
Levin, Peres and Wilmer, Markov Chains and Mixing Times, second edition (standard reference, not scraped)