LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Equivalent event tests for a discrete stopping time

Statement

For a filtration $(F_{n})$ and $τ : Ω \to N_{0} \cup {\infty}$ , the following are equivalent:

${τ \leq n} \in F_{n}$ for every $n$ ;
${τ = n} \in F_{n}$ for every $n$ ;
${τ > n} \in F_{n}$ for every $n$ .

Under any of them, ${τ \geq n} \in F_{n - 1}$ for $n \geq 1$ .

Facts & Assumptions

Given: The hypotheses, objects, and conventions in the Statement.

[F1]

Discrete stopping time is condition (1).

[F2]

Sigma-algebras supplies closure under complements, differences, and finite unions.

Proof

1.1

Conditions (1) and (3) are equivalent because ${τ > n} = Ω ∖ {τ \leq n}$ and $F_{n}$ is closed under complements.

1.2

From (1), ${τ = 0} = {τ \leq 0} \in F_{0}$ and, for $n \geq 1$ , ${τ = n} = {τ \leq n} ∖ {τ \leq n - 1} \in F_{n},$ using $F_{n - 1} \subseteq F_{n}$ . Thus (1) implies (2).

F1F2

1.3

From (2), ${τ \leq n} = ⋃_{k = 0}^{n} {τ = k} \in F_{n},$ since each summand lies in $F_{k} \subseteq F_{n}$ . Thus (2) implies (1), completing the equivalence.

2.1

Finally, for $n \geq 1$ , ${τ \geq n} = Ω ∖ {τ \leq n - 1} \in F_{n - 1} .$ The value $\infty$ is automatically included in this event.

F1F2∎

Depends on

Used by

Wald first equation under integrable stopping Corollary
Integrable stopping time alone does not suffice for arbitrary martingale increments Counterexample
Sigma-algebra at a stopping time Definition
A stopped random variable is measurable at the stopping time Lemma
First hitting time of an adapted process is a stopping time Lemma
Minimum, maximum, and bounded shifts of stopping times Lemma
A stopped martingale is a martingale Theorem

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

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