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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.
A common formulation of the martingale convergence theorem for discrete-time martingales is the following. Let be a supermartingale. Suppose that the supermartingale is bounded in the sense that
where is the negative part of , defined by . Then the sequence converges almost surely to a random variable with finite expectation.
There is a symmetric statement for submartingales with bounded expectation of the positive part. A supermartingale is a stochastic analogue of a non-increasing sequence, and the condition of the theorem is analogous to the condition in the monotone convergence theorem that the sequence be bounded from below. The condition that the martingale is bounded is essential; for example, an unbiased random walk is a martingale but does not converge.
As intuition, there are two reasons why a sequence may fail to converge. It may go off to infinity, or it may oscillate. The boundedness condition prevents the former from happening. The latter is impossible by a "gambling" argument. Specifically, consider a stock market game in which at time , the stock has price . There is no strategy for buying and selling the stock over time, always holding a non-negative amount of stock, which has positive expected profit in this game.
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