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 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.
- 1 Statement for discrete-time martingales
- 2 Statements for the general case
- 3 Doob's upcrossing inequality
- 4 Applications
- 5 See also
- 6 References
Statement for discrete-time martingalesEdit
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. 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 such that always holds. In the event that the sequence does not converge, then and differ. If also the sequence is bounded, then there are some real numbers and such that and the sequence crosses the interval infinitely often. That is, the sequence is eventually less than , and at a later time exceeds , and at an even later time is less than , and so forth ad infinitum. These periods where the sequence starts below and later exceeds are called "upcrossings".
Consider a stock market game in which at time , one may buy or sell shares of the stock at price . On the one hand, it can be shown from the definition of a supermartingale that for any there is no strategy which maintains a non-negative amount of stock and has positive expected profit after playing this game for steps. 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 , and sell it when the price exceeds . Indeed, if is the number of upcrossings in the sequence by time , then the profit at time is at least : each upcrossing provides at least profit, and if the last action was a "buy", then in the worst case the buying price was and the current price is . But any strategy has expected profit at most , so necessarily
By the monotone convergence theorem for expectations, this means that
so the expected number of upcrossings in the whole sequence is finite. It follows that the infinite-crossing event for interval occurs with probability . By a union bound over all rational and , with probability , no interval exists which is crossed infinitely often. If for all there are finitely many upcrossings of interval , then the limit inferior and limit superior of the sequence must agree, so the sequence must converge. This shows that the martingale converges with probability .
Failure of convergence in meanEdit
Under the conditions of the martingale convergence theorem given above, it is not necessarily true that the supermartingale converges in mean (i.e. that ).
As an example, let be a random walk with . Let be the first time when , and let be the stochastic process defined by . Then is a stopping time with respect to the martingale , so is also a martingale, referred to as a stopped martingale. In particular, is a supermartingale which is bounded below, so by the martingale convergence theorem it converges pointwise almost surely to a random variable . But if then , so is almost surely zero.
This means that . However, for every , since is a random walk which starts at and subsequently makes mean-zero moves (alternately, note that since is a martingale). Therefore cannot converge to in mean. Moreover, if were to converge in mean to any random variable , then some subsequence converges to almost surely. So by the above argument almost surely, which contradicts convergence in mean.
Statements for the general caseEdit
Doob's first martingale convergence theoremEdit
Doob's first martingale convergence theorem provides a sufficient condition for the random variables to have a limit as in a pointwise sense, i.e. for each in the sample space individually.
For , let and suppose that
Then the pointwise limit
Doob's second martingale convergence theoremEdit
It is important to note that the convergence in Doob's first martingale convergence theorem is pointwise, not uniform, and is unrelated to convergence in mean square, or indeed in any Lp space. In order to obtain convergence in L1 (i.e., convergence in mean), one requires uniform integrability of the random variables . By Chebyshev's inequality, convergence in L1 implies convergence in probability and convergence in distribution.
The following are equivalent:
- is uniformly integrable, i.e.
Doob's upcrossing inequalityEdit
The following result, called Doob's upcrossing inequality or, sometimes, Doob's upcrossing lemma, is used in proving Doob's martingale convergence theorems. 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.
Let be a natural number. Let be a supermartingale with respect to a filtration . Let , be two real numbers with . Define the random variables so that is the maximum number of disjoint intervals with , such that . These are called upcrossings with respect to interval . Then
Convergence in LpEdit
Let be a continuous martingale such that
for some . Then there exists a random variable such that as both -almost surely and in .
The statement for discrete-time martingales is essentially identical, with the obvious difference that the continuity assumption is no longer necessary.
Lévy's zero–one lawEdit
Doob's martingale convergence theorems imply that conditional expectations also have a convergence property.
both -almost surely and in .
This result is usually called Lévy's zero–one law or Levy's upwards theorem. The reason for the name is that if is an event in , then the theorem says that almost surely, i.e., the limit of the probabilities is 0 or 1. 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. This sounds almost like a tautology, but the result is still non-trivial. For instance, it easily implies Kolmogorov's zero–one law, since it says that for any tail event A, we must have almost surely, hence .
Similarly we have the Levy's downwards theorem :
Let be a probability space and let be a random variable in . Let be any decreasing sequence of sub-sigma algebras of , and define to be the intersection. Then
both -almost surely and in .
This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. (January 2012) (Learn how and when to remove this template message)
- Doob, J. L. (1953). Stochastic Processes. New York: Wiley.
- Durrett, Rick (1996). Probability: theory and examples (Second ed.). Duxbury Press. ISBN 978-0-534-24318-0.; Durrett, Rick (2010). 4th edition. ISBN 9781139491136.
- "Martingale Convergence Theorem" (PDF). Massachusetts Institute of Tecnnology, 6.265/15.070J Lecture 11-Additional Material, Advanced Stochastic Processes, Fall 2013, 10/9/2013.
- Bobrowski, Adam (2005). Functional Analysis for Probability and Stochastic Processe: An Introduction. Cambridge University Press. pp. 113–114. ISBN 9781139443883.
- Gushchin, A. A. (2014). "On pathwise counterparts of Doob's maximal inequalities". Proceedings of the Steklov Institute of Mathematics. 287 (287): 118–121. arXiv:1410.8264. doi:10.1134/S0081543814080070.
- Doob, Joseph L. (2012). Measure theory. Graduate Texts in Mathematics, Vol. 143. Springer. p. 197. ISBN 9781461208778.
- Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1. (See Appendix C)