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A nonnegative martingale converges almost surely

Statement

Assume AC. Every nonnegative martingale $M$ has an almost-sure finite integrable limit $M_{\infty}$ , with $E M_{\infty} \leq E M_{0}$ . Equality and $L^{1}$ convergence are not asserted without uniform integrability.

Facts & Assumptions

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

[F1]

Doob submartingale convergence theorem gives a finite integrable almost-sure limit from bounded positive-part expectations.

[F2]

Fatou's lemma compares its expectation with the constant martingale expectations.

[F3]

The Axiom of Choice states AC, assumed here because F1 and the martingale interface require it.

Proof

1.1

Since $M_{n} \geq 0$ , $M_{n}^{+} = M_{n}$ , and the martingale identity gives $E M_{n} = E M_{0}$ for every $n$ . Thus F1 applies and gives $M_{n} \to M_{\infty}$ almost surely with $M_{\infty}$ finite and integrable.

2.1

Fatou gives $E M_{\infty} \leq {lim inf}_{n} E M_{n} = E M_{0} .$ The inequality can be strict, so neither equality nor $L^{1}$ convergence follows from nonnegativity alone. AC has only the role in F3.

F2F3step 1.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

van der Vaart, Martingales, Diffusions and Financial Mathematics, Exercise 2.22, p. 15 (standard reference, not scraped)