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===== Supermartingale convergence results =====
**__Theorem__**.
Let $\mcf=(\mcf_n)_{n\in\nset}$ be a filtration and let $\seq{X_n}{n\in\nset}$ be an $\mcf$-adapted sequence of $\lone$-random variables such that
* $M=\sup_{n\in\nset} \PE[(X_n)^-]<\infty$
* for all $n\geq 1$, we have $\PE[X_{n}|\mcf_{n-1}]\leq X_{n-1}$.
that is, $\seq{X_n}{n\in\nset}$ is a $(\mcf_n)_{n\in\nset}$-supermartingale, with negative part bounded in $\lone$.
Then, almost surely, $X_\infty=\lim_{n\to\infty} X_n$ exists and is in $\lone$.
==== Proof ====
Let $a
**__Corollary__** Assume that $\mcf=(\mcf_n)_{n\in\nset}$ is a filtration and let $\seq{X_n}{n\in\nset}$ be an $\mcf$-adapted sequence of $\lone$ random variables such that
* $M=\sup_{n\in\nset} \PE[(X_n)^+]<\infty$
* for all $n\geq 1$, we have $\PE[X_{n}|\mcf_{n-1}]\geq X_{n-1}$.
that is, $\seq{X_n}{n\in\nset}$ is a $\mcf$-submartingale, with positive part bounded in $\lone$.
Then, almost surely, $X_\infty=\lim_{n\to\infty} X_n$ exists and is in $\lone$.