Hello,

Suppose $(X_{n}(t))_{n\geq 1}$ is a sequence of real valued stochastic processes, and $T>0$ a fixed number.

Do we have the following implication ?

$\displaystyle{ \lim_{n \to \infty} \sup_{t\in[0,T]}} \mathbb{E}[|X_n(t)|] =0$ implies $\displaystyle{ \lim_{n \to \infty} \mathbb{E}[\sup_{t\in[0,T]}}|X_n(t)|] =0$

If not, what are the weakest conditions on $X_n(t)$ such that the above implication is true ?

Edit 2 : is the implication true if \begin{equation} \mathbb{E}\left[\displaystyle{\sup_{n>0}}\ |X_n(t+h)-X_n(t)|\right]\leq c(h) \end{equation}

with $\displaystyle{\lim_{h\to 0}}\ c(h)=0$

Edit 1 : is the implication true if \begin{equation} \displaystyle{\sup_{n>0}}\ \mathbb{E}\left[|X_n(t+h)-X_n(t)|\right]\leq c(h) \end{equation} with $\displaystyle{\lim_{h\to 0}}\ c(h)=0$. Proven false by Jeff Schenker (cf below).