proof of martingale convergence theorem

Let $(X_{n})_{n\in\mathbb{N}}$ be a supermartingale such that $\mathbb{E}|X_{n}|\leq M$ , and let $a<b$ . We define a random variable counting how many times the process crosses the stripe between $a$ and $b$ :

U_{n}:=\max\{0,r\in\{1,\ldots,n\}\mid\exists 0\leq s_{1}<t_{1}<s_{2}<t_{2}<% \ldots<s_{r}<t_{r}\leq n\;\forall i\in\{1,\ldots,r\}\colon X_{s_{i}}\leq a% \wedge X_{t_{i}}\geq b\}.

Obviously $U_{n+1}\geq U_{n}$ therefore $U_{\infty}=\lim_{n\to\infty}U_{n}$ exists almost surely. Next we will construct a new process that mirrors the movement of $X_{n}$ but only if the original process is underway of going from below $a$ to over $b$ , and is constant otherwise. To do this let $C_{1}:=\chi[X_{0}<a]$ , $C_{k}:=\chi[C_{k-1}=0\wedge X_{k-1}<a]+\chi[C_{k-1}=1\wedge X_{k-1}\leq b]$ for $k\geq 2$ , and define $Y_{0}:=0$ , $Y_{n}:=\sum\limits_{k=1}^{n}C_{k}(X_{k}-X_{k-1})$ . Then $Y_{n}$ is also a supermartingale, and the inequality $Y_{n}\geq(b-a)U_{n}-|X_{n}-a|$ holds, which gives $0\geq\mathbb{E}(Y_{n})\geq(b-a)\mathbb{E}(U_{n})-\mathbb{E}|X_{n}-a|$ . After rearrangement we get

\mathbb{E}(U_{n})\leq\frac{\mathbb{E}|X_{n}-a|}{b-a}\leq\frac{\mathbb{E}|X_{n}% |+|a|}{b-a}\leq\frac{M+|a|}{b-a}.

Therefore by the monotone convergence theorem

\mathbb{E}(U_{\infty})=\lim_{n\to\infty}\mathbb{E}(U_{n})\leq\frac{M+|a|}{b-a}% <\infty,

which means $\mathbb{P}(U_{\infty}=\infty)=0$ . Since $a$ and $b$ were arbitrary $X=\lim_{n\to\infty}X_{n}$ exists almost surely. Now the Fatou lemma gives

\mathbb{E}|X|=\mathbb{E}(\lim_{n\to\infty}|X_{n}|)\leq\liminf_{n\to\infty}% \mathbb{E}|X_{n}|\leq M<\infty.

Thus $X_{n}$ is in fact convergent almost surely, and $X\in L^{1}.\qed$

Title	proof of martingale convergence theorem
Canonical name	ProofOfMartingaleConvergenceTheorem
Date of creation	2013-03-22 18:34:33
Last modified on	2013-03-22 18:34:33
Owner	scineram (4030)
Last modified by	scineram (4030)
Numerical id	4
Author	scineram (4030)
Entry type	Proof
Classification	msc 60F15
Classification	msc 60G44
Classification	msc 60G46
Classification	msc 60G42