ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Uniform integrability repairs the expectation limit

Example

Let $X_{n} \to X$ in probability and suppose $∣ X_{n} ∣ \leq M$ almost surely for one finite constant $M$ and every $n$ . Then $X_{n} \to X$ in $L^{1}$ and $E X_{n} \to E X$ .

Facts & Assumptions

Given: $X_{n} \to X$ in probability and $∣ X_{n} ∣ \leq M$ almost surely for all $n$ .

[L1]

Uniform integrability is the uniform decay of tail integrals (A uniformly integrable family).

[L2]

Under probability convergence, uniform integrability is equivalent to $L^{1}$ convergence (Uniform integrability characterizes $L^{1}$ convergence under probability convergence).

Verification

technique · direct

1.1

If $R > M$ , then $∣ X_{n} ∣ 1_{∣ X_{n} ∣ > R} = 0$ almost surely for every $n$ . Hence the family $(X_{n})$ is uniformly integrable by [L1].

2.1

By [L2], $X_{n} \to X$ in $L^{1}$ . Therefore $∣ E X_{n} - E X ∣ \leq E ∣ X_{n} - X ∣ \to 0$ , proving convergence of expectations.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Rick Durrett, Probability: Theory and Examples, 5th ed., Theorem 4.6.3 (standard reference, not scraped)