Martingale Convergence Theorem

Statement

Let $(X_n)_{n\ge 0}$ be a martingale. If

$$ \sup_{n\ge 0}\mathbb E[X_n^+]<\infty, $$

then there exists a random variable $X_\infty$ such that

$$ X_n\to X_\infty \qquad \text{almost surely.} $$

If $(X_n)$ is a nonnegative martingale, then

$$ X_n^+=X_n $$

and

$$ \mathbb E[X_n]=\mathbb E[X_0]. $$

So the condition holds automatically, and every nonnegative martingale converges almost surely.

Almost sure convergence does not by itself imply

$$ \mathbb E[X_\infty]=\mathbb E[X_0]. $$

That extra conclusion requires uniform integrability.