Doob's martingale convergence theorems

In mathematics – specifically, in the  – 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.

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 < +∞,

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 P-almost all ω ∈ Ω.

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:

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 P-almost surely and in L^p(Ω, P; R).

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