TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Uniformly integrable martingale convergence

Statement

Assume AC. A uniformly integrable martingale $M$ converges almost surely and in $L^{1}$ to an integrable random variable $M_{\infty}$ .

Facts & Assumptions

Given: The hypotheses, objects, and conventions in the Statement.

[F1]

A uniformly integrable family implies uniform $L^{1}$ boundedness.

[F2]

Doob submartingale convergence theorem gives an integrable almost-sure limit under the resulting positive-part bound.

[F3]

Almost-sure convergence implies convergence in probability and Uniform integrability plus convergence in probability implies $L^{1}$ convergence upgrade convergence to $L^{1}$ .

[F4]

The Axiom of Choice is inherited from F2's martingale conditional expectations and their countable representatives.

Proof

1.1F1F2

Uniform integrability implies $\sup_{n} E ∣ M_{n} ∣ < \infty$ by F1, hence $\sup_{n} E M_{n}^{+} < \infty$ . Apply F2 to obtain a finite integrable $M_{\infty}$ with $M_{n} \to M_{\infty}$ almost surely.

2.1F3F4step 1.1∎

Almost-sure convergence implies convergence in probability. The family ${M_{n} : n \geq 0}$ is uniformly integrable by hypothesis, so F3 yields $E ∣ M_{n} - M_{\infty} ∣ \to 0$ . Mere $L^{1}$ boundedness was not substituted for uniform integrability. AC is used exactly as described in F4.

Depends on

Used by

Closed martingale characterization Theorem

Dependency tree · two levels

20 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

van der Vaart, Martingales, Diffusions and Financial Mathematics, Theorem 2.23 and proof, pp. 15–16 (standard reference, not scraped)