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I read the proof of $L^p$ convergence theorem of martingales but I can't exactly figure out that whether we can find a square integrable martingale that converges almost surely but not in $L^2$.

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    If you are assuming $\sup_t E[|M_t|^p]<\infty$, for some $p>1$, then the convergence is both in $L^p$ and a.s. The only tricky case is $L^1$2012-11-29
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    No. I mean that is there a case where each ${X_n}^2$ is integrable but the whole thing (which although has an $a.s.$ convergence) does not converge in $L^2$ (in fact as you say we are looking for a martingale which $sup E|X_n|$ is not bounded but $E|X_n|$ is.)2012-11-29

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