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.
[1] 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.
Then the sequence converges almost surely to a random variable
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
As intuition, there are two reasons why a sequence may fail to converge.
Specifically, consider a stock market game in which at time
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.
The reason is that at each time the expected change in stock price, given all past information, is at most zero (by definition of a supermartingale).
But if the prices were to oscillate without converging, then there would be a strategy with positive expected profit: loosely, buy low and sell high.
This argument can be made rigorous to prove the result.
The proof is simplified by making the (stronger) assumption that the supermartingale is uniformly bounded; that is, there is a constant
, one may buy or sell shares of the stock at price
there is no strategy which maintains a non-negative amount of stock and has positive expected profit after playing this game for
On the other hand, if the prices cross a fixed interval
very often, then the following strategy seems to do well: buy the stock when the price drops below
, so necessarily By the monotone convergence theorem for expectations, this means that so the expected number of upcrossings in the whole sequence is finite.
Under the conditions of the martingale convergence theorem given above, it is not necessarily true that the supermartingale
and subsequently makes mean-zero moves (alternately, note that
, Doob's first martingale convergence theorem provides a sufficient condition for the random variables
and suppose that Then the pointwise limit exists and is finite for
In order to obtain convergence in L1 (i.e., convergence in mean), one requires uniform integrability of the random variables
[3] A "gambling" argument shows that for uniformly bounded supermartingales, the number of upcrossings is bounded; the upcrossing lemma generalizes this argument to supermartingales with bounded expectation of their negative parts.
The statement for discrete-time martingales is essentially identical, with the obvious difference that the continuity assumption is no longer necessary.
This result is usually called Lévy's zero–one law or Levy's upwards theorem.
In plain language, if we are learning gradually all the information that determines the outcome of an event, then we will become gradually certain what the outcome will be.
For instance, it easily implies Kolmogorov's zero–one law, since it says that for any tail event A, we must have