Doob's Martingale Convergence Theorem states that a martingale, or more generally a supermartingale, whose expected absolute value stays bounded across time converges almost surely to a finite limit. Named for Joseph Doob, it is a central result of the theory of stochastic processes, guaranteeing that a broad class of random processes that fluctuate without a fixed trend still settle down in the long run.
Facts
StatementAny supermartingale whose expected absolute value stays bounded over time converges almost surely to a finite limit, with a symmetric statement for submartingales bounded above. 1 Classification
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Sources
1. Doob's Martingale Convergence Theorem (Wikipedia)
Wikimedia Foundationlead section, second sentenceQuote, lead section, second sentence
Informally, the martingale convergence theorem typically refers to the result that any supermartingale satisfying a certain boundedness condition must converge.
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