Timeline for expectation of supremum

9 events

when toggle format	what		by	license	comment
Sep 28, 2011 at 16:06	history	edited	user16215	CC BY-SA 3.0	added 253 characters in body
Sep 28, 2011 at 15:18	answer	added	Jeff Schenker		timeline score: 2
Sep 28, 2011 at 14:07	comment	added	kaleidoscop		I don't know if it can helps, but I imagine your assumption with $c(h)$ implies that the family $\{X_n\}$ is tight for the uniform convergence topology, meaning the law of $X_n$ converges weakly for this topology to the law of some random continuous process $X$.
Sep 28, 2011 at 13:34	history	edited	user16215	CC BY-SA 3.0	added 51 characters in body
Sep 28, 2011 at 13:01	history	edited	user16215	CC BY-SA 3.0	added 142 characters in body
Sep 28, 2011 at 12:59	comment	added	user16215		@Anthony Quas : Thank you for this nice counter-example. Do you think the implication is still false if one assumes that $X_n(t)$ is continuous on $[0,T]$, with a modulus of continuity independent of $n$?
Sep 28, 2011 at 12:05	comment	added	The Bridge		Hi I wonder if for some (good) local martingales the implication can be proved using BDG inequlities. Regards
Sep 28, 2011 at 11:57	comment	added	Anthony Quas		The implication is false: Let $X_n(t)$ be a stochastic process on $[0,1]$ with $X_n(t)=1$ for an interval $[(i-1)/2^n,i/2^n]$ where $i$ is chosen uniformly at random from the set $\{1,\ldots,2^n\}$. Then for any fixed $t$, $X_n(t)$ is 1 with probability $2^{-n}$ so that the left side is $2^{-n}$. On the other hand, the right side is 1.
Sep 28, 2011 at 9:57	history	asked	user16215	CC BY-SA 3.0