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

Suppose further that : \begin{equation} \mathbb{E}\left[\displaystyle{\sup_{n>0}}\ |X_n(t+h)-X_n(t)|\right]\leq c(h) \end{equation}

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

Do we have the following implication ?

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

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Have a look here – TheBridge Oct 11 '11 at 9:34
@TheBridge : thanks but there is no satisfactory answer so far – mellow Oct 11 '11 at 14:08

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