Uniformly integrable martingale convergence theorem

A uniformly integrable martingale $M_n$ has almost sure convergence and convergence in L1 to an integrable random variable $M_{\infty }$, and $M_n=\mathbb{E}[M_{\infty }\mid {\mathcal{F}}_n]$.

Stopping a uniformly integrable discrete-time martingale at any stopping time preserves uniform integrability, including when the stopping time can be infinite. For $C={\sup }_n\mathbb{E}|X_n|$, the optional stopping theorem and the Markov inequality giveChoose $R$ first, then $K$, to make this uniformly small.