Doob's martingale convergence theorems

Remarks

The proof of part 1 uses Doob's upcrossing inequality. The bound on $E[X_n^+]$ makes the number of upcrossings of each rational interval finite almost surely, so the limit exists in $[-\infty, \infty]$. Fatou's lemma shows that the limit is finite and integrable. Part 2 follows from part 1 applied to $-X_n$, and part 3 uses that uniform integrability upgrades almost sure convergence to $L^1$ convergence. Example: for independent $\xi_i$ with $P(\xi_i = 0) = P(\xi_i = 2) = 1/2$, the product $M_n = \xi_1 \cdots \xi_n$ is a non-negative martingale with $E[M_n] = 1$. It converges almost surely to $0$ but not in $L^1$, so it is not uniformly integrable, which shows that part 3 needs its hypothesis. The bound in part 1 is needed: the simple symmetric random walk is a martingale with $E[\abs{S_n}] \to \infty$, and it does not converge. The same statements hold in continuous time for submartingales with right-continuous paths.

Going deeper

For a student

These theorems are the main reason martingales are useful in analysis and probability. They give short proofs of the strong law of large numbers, of Lévy's upward theorem and of Kolmogorov's zero–one law. They also give almost sure convergence of likelihood ratios, of branching process normalisations and of the Radon–Nikodym derivatives on finer and finer partitions. The $L^p$ version, for $p > 1$, says that a martingale bounded in $L^p$ converges almost surely and in $L^p$, by Doob's martingale inequality.