# 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. 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}

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}, $C_{k}:=\chi[C_{k-1}=0\wedge X_{k-1} 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 ProofOfMartingaleConvergenceTheorem 2013-03-22 18:34:33 2013-03-22 18:34:33 scineram (4030) scineram (4030) 4 scineram (4030) Proof msc 60F15 msc 60G44 msc 60G46 msc 60G42