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.

Stopped random variable and stopped process

Definition

For an adapted real process $X$ , a stopping time $τ$ , and a fixed real cemetery value $x_{\infty}$ , define $X_{τ} = \sum_{k \geq 0} X_{k} 1_{{τ = k}} + x_{\infty} 1_{{τ = \infty}} .$ The level events are disjoint and exactly one displayed term is active, so the value is unambiguous. If an actual terminal variable $X_{\infty}$ is separately supplied, it may replace $x_{\infty}$ on ${τ = \infty}$ , but that convention must be stated.

The stopped process is $X_{n}^{τ} = X_{n \land τ} .$ Because $n \land τ \in {0, \dots, n}$ , this expression never evaluates the cemetery value and equals the finite sum $\sum_{k = 0}^{n - 1} X_{k} 1_{{τ = k}} + X_{n} 1_{{τ \geq n}} .$

Depends on

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

van der Vaart, Martingales, Diffusions and Financial Mathematics, §§2.3 and 2.8 (standard reference, not scraped)