DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Almost-sure convergence of real random variables

Definition

Let $(X_{n})$ and $X$ be real random variables on one probability space. Write $X_{n} \to X$ almost surely when $P ({ω : X_{n} (ω) \to X (ω) in R}) = 1.$ The convergence in the event is the real convergence of Limits and Cauchy sequences of reals. The set is measurable by The almost-sure convergence event is measurable ↗, so its probability is defined.

Depends on

Used by

Almost-sure convergence does not imply convergence of expectations Counterexample
Almost-sure convergence need not imply Lᵖ convergence Counterexample
Convergence in probability need not be almost sure Counterexample
Lᵖ convergence need not imply almost-sure convergence Counterexample
The almost-sure convergence event is measurable Lemma
Almost-sure convergence implies convergence in probability Theorem
An almost-surely convergent subsequence from convergence in probability Theorem
Dominated convergence in Lᵖ Theorem
Limits in probability are unique almost surely Theorem

Dependency tree · two levels

9 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. Roch, Lecture 3: Modes of convergence, Definition 3.1 (standard reference, not scraped)