There are several convergence theorems for martingales, which follow from Doob's upcrossing lemma. The following says that any $L^1$ -bounded martingale $X_n$ in discrete time converges almost surely. Note that almost-sure convergence (i.e. convergence with probability one) is quite strong, implying the weaker property of convergence in probability. Here, a martingale $(X_n)_{n\in\mathbb{N}}$ is understood to be defined with respect to a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and filtration $(\mathcal{F}_n)_{n\in\mathbb{N}}$ .

The condition that $X_n$ is $L^1$ -bounded is automatically satisfied in many cases. In particular, if $X$ is a non-negative supermartingale then $\mathbb{E}[|X_n|]=\mathbb{E}[X_n]\le\mathbb{E}[X_1]$ for all $n\ge 1$ , so $\mathbb{E}[|X_n|]$ is bounded, giving the following corollary.

As an example application of the martingale convergence theorem, it is easy to show that a standard random walk started started at $0$ will visit every level with probability one.

Proof. Without loss of generality, suppose that $a\le 0$ . Let $T:\Omega\rightarrow\mathbb{N}\cup\{\infty\}$ be the first time $n$ for which $X_n=a$ . It is easy to see that the stopped process $X^T_n$ defined by $X^T_n=X_{\min(n,T)}$ is a martingale and $X^T-a$ is non-negative. Therefore, by the martingale convergence theorem, the limit $X^T_\infty=\lim_{n\rightarrow\infty}X^T_n$ exists and is finite (almost surely). In particular, $|X^T_{n+1}-X^T_n|$ converges to $0$ and must be less than $1$ for large $n$ . However, $|X^T_{n+1}-X^T_n|=1$ whenever $n<T$ , so we have $T<\infty$ and therefore $X_n=a$ for some $n$ .