Doob's martingale convergence theorems

In mathematics – specifically, in the theory of stochastic processes – Doob's martingale convergence theorems are a collection of results on the long-time limits of supermartingales, named after the American mathematician Joseph L. Doob. In mathematics – specifically, in the theory of stochastic processes – Doob's martingale convergence theorems are a collection of results on the long-time limits of supermartingales, named after the American mathematician Joseph L. Doob. In the following, (Ω, F, F∗, P), F∗ = (Ft)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 < +∞, Doob's first martingale convergence theorem provides a sufficient condition for the random variables Nt to have a limit as t → +∞ in a pointwise sense, i.e. for each ω in the sample space Ω individually. For t ≥ 0, let Nt− = max(−Nt, 0) and suppose that