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

###### Theorem (Doob’s Forward Convergence Theorem).

Let $(X_{n})_{n\in\mathbb{N}}$ be a martingale (or submartingale, or supermartingale) such that $\mathbb{E}[|X_{n}|]$ is bounded over all $n\in\mathbb{N}$. Then, with probability one, the limit $X_{\infty}=\lim_{n\rightarrow\infty}X_{n}$ 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 $\mathbb{E}[|X_{n}|]=\mathbb{E}[X_{n}]\leq\mathbb{E}[X_{1}]$ for all $n\geq 1$, so $\mathbb{E}[|X_{n}|]$ is bounded, giving the following corollary.

###### Corollary.

Let $(X_{n})_{n\in\mathbb{N}}$ be a non-negative martingale (or supermartingale). Then, with probability one, the limit $X_{\infty}=\lim_{n\rightarrow\infty}X_{n}$ 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 $(X_{n})_{n\in\mathbb{N}}$ be a standard random walk. That is, $X_{1}=0$ and

 $\mathbb{P}(X_{n+1}=X_{n}+1\mid\mathcal{F}_{n})=\mathbb{P}(X_{n+1}=X_{n}-1\mid% \mathcal{F}_{n})=1/2.$

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

###### Proof.

Without loss of generality, suppose that $a\leq 0$. 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, so we have $T<\infty$ and therefore $X_{n}=a$ for some $n$. ∎

 Title martingale convergence theorem Canonical name MartingaleConvergenceTheorem Date of creation 2013-03-22 18:33:47 Last modified on 2013-03-22 18:33:47 Owner gel (22282) Last modified by gel (22282) Numerical id 5 Author gel (22282) Entry type Theorem Classification msc 60G46 Classification msc 60G44 Classification msc 60G42 Classification msc 60F15 Related topic Martingale Related topic MartingaleProofOfKolmogorovsStrongLawForSquareIntegrableVariables Related topic MartingaleProofOfTheRadonNikodymTheorem Related topic UpcrossingsAndDowncrossings