Stable convergence implies convergence of second moments?

probability probability-theory probability-limit-theorems

Solution 1:

Notice that taking $U=1$ gives that $X_n \to X$ in distribution. Using Skorokhod's representation theorem, we can assume that there is almost sure convergence.

Now since $X_n$ is bounded in every $L^p$, $X_n^2$ is uniformly integrable. Thus Vitali's theorem gives the convergence $X_n \to X$ in $L^2$.

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