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Concept
Doob's martingale convergence theorems
Summary
In mathematics specifically, in the theory of stochastic processes Doob's martingale convergence theorems are a collection of results on the limits of supermartingales, named after the American mathematician Joseph L. Doob. Informally, the martingale convergence theorem typically refers to the result that any supermartingale satisfying a certain boundedness condition must converge. One may think of supermartingales as the random variable analogues of non-increasing sequences; from this perspective, the martingale convergence theorem is a random variable analogue of the monotone convergence theorem, which states that any bounded monotone sequence converges. There are symmetric results for submartingales, which are analogous to non-decreasing sequences.
Statement for discrete-time martingales
A common formulation of the martingale convergence theorem for discrete-time martingales is the following. Let X_1, X_2, X_3, \dots be a supermartingale. Suppose t
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