TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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A mean-one likelihood-ratio limit implies contiguity

Statement

For each $n$ , let $P_{n}, Q_{n}$ be probability measures on one measurable space, and let $L_{n} \geq 0$ satisfy $Q_{n} (A) = \int_{A} L_{n} d P_{n}$ for every measurable $A$ . Suppose $E_{P_{n}} L_{n} = 1$ , $L_{n} \Rightarrow L$ under $P_{n}$ , $L \geq 0$ , and $E L = 1$ . Then $(Q_{n})$ is contiguous with respect to $(P_{n})$ .

Facts & Assumptions

Given: Probability measures $P_{n}, Q_{n}$ , mean-one densities $L_{n}$ , and the stated nonnegative distributional limit $L$ with matching mean.

[F1]

Contiguous sequences of probability measures: $Q_{n} ◃ P_{n}$ means $P_{n} (A_{n}) \to 0$ implies $Q_{n} (A_{n}) \to 0$ for every event sequence $A_{n}$ .

[F2]

Convergence in distribution of random elements: $L_{n} \Rightarrow L$ means expectations converge for every bounded continuous function.

Proof

technique · direct uniform-integrability argument

1.1F2givenalgebra

For fixed $M > 0$ , $x \mapsto x \land M$ is bounded and continuous on $[0, \infty)$ , so [F2] and the mean assumptions give $E (L_{n} - M)_{+} = E L_{n} - E (L_{n} \land M) ⟶ E L - E (L \land M) = E (L - M)_{+} .$

2.1step 1.1givenalgebra

If $x > 2 M$ , then $x \leq 2 (x - M)$ ; hence ${lim sup}_{n} E [L_{n} 1_{{L_{n} > 2 M}}] \leq 2 E (L - M)_{+} .$ The right side tends to zero as $M \to \infty$ because $L$ is integrable. Each of the finitely many initial integrable $L_{n}$ also has vanishing tail integral, so $(L_{n})$ is uniformly integrable.

3.1F1F2step 2.1givenalgebra∎

For any event sequence $A_{n}$ and any $M > 0$ , $Q_{n} (A_{n}) = E [L_{n} 1_{A_{n}}] \leq M P_{n} (A_{n}) + E [L_{n} 1_{{L_{n} > M}}] .$ If $P_{n} (A_{n}) \to 0$ , the first term tends to zero for fixed $M$ , and uniform integrability makes the limsup of the second term arbitrarily small as $M \to \infty$ . Thus $Q_{n} (A_{n}) \to 0$ , which is $Q_{n} ◃ P_{n}$ by [F1].

Depends on

Used by

Binary laws with a rare event are not contiguous Counterexample
Finite-prior Bayes envelopes converge under pointwise LAN Lemma
Le Cam's third lemma for a Gaussian likelihood-ratio limit Theorem

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

Peter Bartlett, Theoretical Statistics, Stat 210B Lecture 21, PDF pp. 12–15 (standard reference, not scraped)
John Duchi, A few notes on contiguity, asymptotics, and local asymptotic normality, §§1.2–1.3 (standard reference, not scraped)