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.

Independent random elements

Definition

Let $X_{i} : (Ω, F, P) \to (S_{i}, Σ_{i})$ be random elements on a common probability space. For each $i$ , write

$σ (X_{i}) : = {X_{i}^{- 1} (B) : B \in Σ_{i}} \subseteq F .$

The family $(X_{i})_{i \in I}$ is independent when the sigma-algebras $(σ (X_{i}))_{i \in I}$ are independent in the sense of Independent sigma-algebras and independent events.

When $Σ_{i}$ is a Borel sigma-algebra, this agrees with the notation $σ (X_{i})$ from The sigma-algebra generated by a function.

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, Definitions 3.2 and 3.4 (standard reference, not scraped)
Rick Durrett, Probability: Theory and Examples, 5th ed., Section 2.1 (standard reference, not scraped)