When are the limits of Martingales are Martingales? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T05:56:24Z http://mathoverflow.net/feeds/question/83304 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83304/when-are-the-limits-of-martingales-are-martingales When are the limits of Martingales are Martingales? Ben 2011-12-13T01:35:16Z 2011-12-13T01:35:16Z <p>Suppose I have a sequence of continuous time random variables $X_n(t)$ where $t \in [0,1]$, adapted to a filtration $F_t$, that are martingales with respect to this filtration and that $\sup_n E[\sup_{0 \leq t \leq 1 } |X_n(t)|] &lt; \infty$. Note that the filtration does not depend on $n$. I assume quite strong convergence of the $X_n(t)$ to a limit $X(t)$, namely that they converge uniformly on $[0,1]$ and for each $t$, $X_n(t)$ converges in $L^1$. I would like to know what are the weakest conditions I can assume about the $X_n(t)$ so that $X(t)$ will also be a martingale. If I assume the $X_n(t)$ are continuous or even left continuous then I believe that the limit will also be a martingale. If I just want to assume right continuity, I think I need that the fixed time distributions of $X(t)$ are non-singular. What I'd really like to know is if some kind of continuity of the $X_n$'s is a requirement to push the martingale through to the limit. </p>