Under which condition does $E[X_n^2]\rightarrow E[X^2]$ impliy that $X_n\xrightarrow{L^2}X$. I think it is sufficient that $X_n$ is uniformly integrable but can't remember a proof at the moment. So is my guess correct and if not are there other conditions that satisfy this implication?
2026-03-27 02:39:33.1774579173
When does Moment Convergence imply convergence in $L^p$?
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