martingale convergence theorem

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 $(\mathrm{\Omega },ℱ,\mathbb{P})$ and filtration ${({ℱ}_n)}_{n\in \mathbb{N}}$.

Theorem (Doob’s Forward Convergence Theorem).

Let ${\mathrm{(}{X}_n\mathrm{)}}_{n\mathrm{\in }\mathrm{N}}$ be a martingale (or submartingale, or supermartingale) such that $\mathrm{E}\mathit{}\mathrm{[}\mathrm{|}{X}_n\mathrm{|}\mathrm{]}$ is bounded over all $n\mathrm{\in }\mathrm{N}$. Then, with probability one, the limit $X_{\mathrm{\infty }}\mathrm{=}{\lim }_{n\mathrm{\to }\mathrm{\infty }}\mathit{}{X}_n$ exists and is finite.

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 $𝔼[|X_n|]=𝔼[X_n]\le 𝔼[X_1]$ for all $n\ge 1$, so $𝔼[|X_n|]$ is bounded, giving the following corollary.

Corollary.

Let ${\mathrm{(}{X}_n\mathrm{)}}_{n\mathrm{\in }\mathrm{N}}$ be a non-negative martingale (or supermartingale). Then, with probability one, the limit $X_{\mathrm{\infty }}\mathrm{=}{\lim }_{n\mathrm{\to }\mathrm{\infty }}\mathit{}{X}_n$ exists and is finite.

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.

Corollary.

Let ${\mathrm{(}{X}_n\mathrm{)}}_{n\mathrm{\in }\mathrm{N}}$ be a standard random walk. That is, $X_{\mathrm{1}}\mathrm{=}\mathrm{0}$ and

$$\mathbb{P}(X_{n+1}=X_n+1\mid {ℱ}_n)=\mathbb{P}(X_{n+1}=X_n-1\mid {ℱ}_n)=1/2.$$

Then, for every integer $a$, with probability one $X_n\mathrm{=}a$ for some $n$.

Proof.

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