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 longtime limits of supermartingales, named after the American mathematician Joseph L. Doob.^{[1]}
Contents
Statement of the theorems[edit]
In the following, (Ω, F, F_{∗}, P), F_{∗} = (F_{t})_{t ≥ 0}, will be a filtered probability space and N : [0, +∞) × Ω → R will be a rightcontinuous supermartingale with respect to the filtration F_{∗}; in other words, for all 0 ≤ s ≤ t < +∞,
Doob's first martingale convergence theorem[edit]
Doob's first martingale convergence theorem provides a sufficient condition for the random variables N_{t} 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 N_{t}^{−} = max(−N_{t}, 0) and suppose that
Then the pointwise limit
exists and is finite for Palmost all ω ∈ Ω.^{[2]}
Doob's second martingale convergence theorem[edit]
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 L^{p} space. In order to obtain convergence in L^{1} (i.e., convergence in mean), one requires uniform integrability of the random variables N_{t}. By Chebyshev's inequality, convergence in L^{1} implies convergence in probability and convergence in distribution.
The following are equivalent:
 (N_{t})_{t > 0} is uniformly integrable, i.e.
 there exists an integrable random variable N ∈ L^{1}(Ω, P; R) such that N_{t} → N as t → +∞ both Palmost surely and in L^{1}(Ω, P; R), i.e.
Corollary: convergence theorem for continuous martingales[edit]
Let M : [0, +∞) × Ω → R be a continuous martingale such that
for some p > 1. Then there exists a random variable M ∈ L^{p}(Ω, P; R) such that M_{t} → M as t → +∞ both Palmost surely and in L^{p}(Ω, P; R).
Discretetime results[edit]
Similar results can be obtained for discretetime supermartingales and submartingales, the obvious difference being that no continuity assumptions are required. For example, the result above becomes
Let M : N × Ω → R be a discretetime martingale such that
for some p > 1. Then there exists a random variable M ∈ L^{p}(Ω, P; R) such that M_{k} → M as k → +∞ both Palmost surely and in L^{p}(Ω, P; R)
Convergence of conditional expectations: Lévy's zero–one law[edit]
Doob's martingale convergence theorems imply that conditional expectations also have a convergence property.
Let (Ω, F, P) be a probability space and let X be a random variable in L^{1}. Let F_{∗} = (F_{k})_{k∈N} be any filtration of F, and define F_{∞} to be the minimal σalgebra generated by (F_{k})_{k∈N}. Then
both Palmost surely and in L^{1}.
This result is usually called Lévy's zero–one law or Levy's upwards theorem. The reason for the name is that if A is an event in F_{∞}, 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 nontrivial. 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 (Ω, F, P) be a probability space and let X be a random variable in L^{1}. Let (F_{k})_{k∈N} be any decreasing sequence of subsigma algebras of F, and define F_{∞} to be the intersection. Then
both Palmost surely and in L^{1}.
Doob's upcrossing inequality[edit]
The following result, called Doob's upcrossing inequality or, sometimes, Doob's upcrossing lemma, is used in proving Doob's martingale convergence theorems.^{[2]}
Hypothesis.
Let be a natural number. Let , for , be a martingale with respect to a filtration , for . Let , be two real numbers with .
Define the random variables , for , as follows: if and only if is the largest integer such that there exist integers , satisfying 1 ≤ < and, for , for each pair the inequalities and are satisfied. Each is called the number of upcrossings with respect to the interval for the martingale , .
Conclusion.
 ^{[3]}^{[4]}
See also[edit]
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References[edit]
 ^ Doob, J. L. (1953). Stochastic Processes. New York: Wiley.
 ^ ^{a} ^{b} "Martingale Convergence Theorem" (PDF). Massachusetts Institute of Tecnnology, 6.265/15.070J Lecture 11Additional 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 3540047581. (See Appendix C)
 Durrett, Rick (1996). Probability: theory and examples (Second ed.). Duxbury Press. ISBN 9780534243180.; Durrett, Rick (2010). 4th edition. ISBN 9781139491136.