DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Contiguous sequences of probability measures

Definition

Let $(X, A)$ be one measurable space and let $P_{n}, Q_{n}$ be probability measures on it (Probability measures and probability spaces). The sequence $(Q_{n})$ is contiguous with respect to $(P_{n})$ , written $Q_{n} ◃ P_{n}$ , if for every sequence $A_{n} \in A$ , $P_{n} (A_{n}) ⟶ 0 ⟹ Q_{n} (A_{n}) ⟶ 0.$ The sequences are mutually contiguous, written $Q_{n} ◃ ▹ P_{n}$ , if both $Q_{n} ◃ P_{n}$ and $P_{n} ◃ Q_{n}$ .

Contiguity is a relation between sequences of laws, not absolute continuity at each finite $n$ . It says that events negligible under one sequence remain negligible under the other in the indicated direction.

Depends on

Probability measures and probability spaces

Used by

Binary laws with a rare event are not contiguous Counterexample
Local asymptotic normality for local experiments Definition
A mean-one likelihood-ratio limit implies contiguity Theorem
Le Cam's third lemma for a Gaussian likelihood-ratio limit Theorem

Dependency tree · two levels

3 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

Peter Bartlett, Theoretical Statistics, Stat 210B Lecture 21, PDF pp. 8–14 (standard reference, not scraped)
John Duchi, A few notes on contiguity, asymptotics, and local asymptotic normality, §§1.1–1.3 (standard reference, not scraped)