DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Discrete stopping time

Definition

For a filtration $(F_{n})_{n \geq 0}$ , a map $τ : Ω ⟶ N_{0} \cup {\infty}$ is a stopping time if ${τ \leq n} \in F_{n}$ for every $n \geq 0$ . It is bounded if some deterministic $N$ satisfies $τ \leq N$ almost surely, and finite if $P (τ < \infty) = 1$ . These notions are distinct: finite does not mean bounded.

The event condition is literal, not merely modulo null sets; no completion of the filtration is assumed. Measurability as an extended-integer-valued random variable follows from the equivalent event tests below.

Depends on

Used by

Sequential likelihood-ratio test Algorithm
A last exit time need not be a stopping time Counterexample
Anytime-valid p-value processes Definition
Hitting, return, and visit times Definition
Sigma-algebra at a stopping time Definition
Stopped random variable and stopped process Definition
A data-dependent threshold crossing Example
Repeated unadjusted tests inflate global-null error Example
Equivalent event tests for a discrete stopping time Lemma
Finite-horizon crossing bound for a nonnegative test supermartingale Lemma
Minimum, maximum, and bounded shifts of stopping times Lemma
Discrete strong Markov property Theorem
Discrete Ville inequality and anytime threshold validity Theorem
Renewal decomposition at successive returns Theorem
Terminal error bounds for a sequential likelihood-ratio test Theorem
Ville's inequality for nonnegative test supermartingales 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

van der Vaart, Martingales, Diffusions and Financial Mathematics, §2.3, pp. 9–11 (standard reference, not scraped)