DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Tail sigma-algebra of a sequence

Definition

Let $(X_{n})_{n \in N}$ be random elements on a common probability space. For $m \in N$ , write

$σ (X_{n} : n \geq m) : = σ_{Ω} (⋃_{n \geq m} σ (X_{n})),$

the sigma-algebra generated by all coordinate sigma-algebras from time $m$ onward.

The tail sigma-algebra of the sequence is

$T (X_{n} : n \in N) : = ⋂_{m \in N} σ (X_{n} : n \geq m) .$

Thus a tail event is one whose membership is unchanged by altering only finitely many initial coordinates.

Depends on

Used by

Dependency tree · two levels

8 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

S. R. S. Varadhan, Probability Theory, discussion before Theorem 3.15 (standard reference, not scraped)