It seems that most 0-1 laws are stated in terms of almost sure convergence. I am aware that the martingale convergence theorem gives us $L^1$ convergence if the sequence of random variables is uniformly integrable. But it is still silent on $L^p$ convergence for $p>2$. Since $L^p$ convergence and almost sure convergence are different, I am wondering if there is an analogue of 0-1 law (or any asymptotic convergence theorem) for $L^p$ norms. Thanks.
Is there an analog of 0-1 law for $L^p$ convergence?
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probability-theory