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Doob's martingale convergence theorems : ウィキペディア英語版 | Doob's martingale convergence theorems In mathematicsspecifically, in stochastic analysisDoob's martingale convergence theorems are a collection of results on the long-time limits of supermartingales, named after the American mathematician Joseph L. Doob. ==Statement of the theorems== In the following, (Ω, ''F'', ''F''∗, P), ''F''∗ = (''F''''t'')''t''≥0, will be a filtered probability space and ''N'' : [0, +∞) × Ω → R will be a right-continuous supermartingale with respect to the filtration ''F''∗; in other words, for all 0 ≤ ''s'' ≤ ''t'' < +∞, :
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