Doob's martingale convergence theorems

In mathematics – specifically, in stochastic analysis – Doob's martingale convergence theorems are a collection of results on the long-time limits of supermartingales, named after the American mathematician Joseph L. Doob.

Statement of the theorems

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 right-continuous supermartingale with respect to the filtration F_∗; in other words, for all 0 ≤ s ≤ t < +∞,

N_{s} \geq \mathbf{E} \big[ N_{t} \big| F_{s} \big].

Doob's first martingale convergence theorem

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

\sup_{t > 0} \mathbf{E} \big[ N_{t}^{-} \big] < + \infty.

Then the pointwise limit

N(\omega) = \lim_{t \to + \infty} N_{t} (\omega)

exists finite for P-almost all ω ∈ Ω.

Doob's second martingale convergence theorem

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 L¹ (i.e., convergence in mean), one requires uniform integrability of the random variables N_t. By Chebyshev's inequality, convergence in L¹ implies convergence in probability and convergence in distribution.

The following are equivalent:

(N_t)_t>0 is uniformly integrable, i.e.

\lim_{C \to \infty} \sup_{t > 0} \int_{\{ \omega \in \Omega | N_{t} (\omega) > C \}} \big| N_{t} (\omega) \big| \, \mathrm{d} \mathbf{P} (\omega) = 0;

there exists an integrable random variable N ∈ L¹(Ω, P; R) such that N_t → N as t → +∞ both P-almost surely and in L¹(Ω, P; R), i.e.

\mathbf{E} \big[ \big| N_{t} - N \big| \big] = \int_{\Omega} \big| N_{t} (\omega) - N (\omega) \big| \, \mathrm{d} \mathbf{P} (\omega) \to 0 \mbox{ as } t \to + \infty.

Corollary: convergence theorem for continuous martingales

Let M : [0, +∞) × Ω → R be a continuous martingale such that

\sup_{t > 0} \mathbf{E} \big[ \big| M_{t} \big|^{p} \big] < + \infty

for some p > 1. Then there exists a random variable M ∈ L^p(Ω, P; R) such that M_t → M as t → +∞ both P-almost surely and in L^p(Ω, P; R).

Discrete-time results

Similar results can be obtained for discrete-time 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 discrete-time martingale such that

\sup_{k \in \mathbf{N}} \mathbf{E} \big[ \big| M_{k} \big|^{p} \big] < + \infty

for some p > 1. Then there exists a random variable M ∈ L^p(Ω, P; R) such that M_k → M as k → +∞ both P-almost surely and in L^p(Ω, P; R)

Convergence of conditional expectations: Lévy's zero–one law

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 L¹. Let F_∗ = (F_k)_k∈N be any filtration of F, and define F_∞ to be the minimal σ-algebra generated by (F_k)_k∈N. Then

\mathbf{E} \big[ X \big| F_{k} \big] \to \mathbf{E} \big[ X \big| F_{\infty} \big] \mbox{ as } k \to \infty

both P-almost surely and in L¹.

This result is usually called Lévy's zero–one law. The reason for the name is that if A is an event in F_∞, then the theorem says that $\mathbf{P}[ A | F_{k} ] \to \mathbf{1}_A$ 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 $\mathbf{P}[ A ] = \mathbf{1}_A$ almost surely, hence $\mathbf{P}[ A ]\in\{0,1\}$ .

References

Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1. (See Appendix C)

Durrett, Rick (1996). Probability: theory and examples (Second ed.). Duxbury Press. ISBN 978-0-534-24318-0.