The answer seems to obviously be yes. But it's not so obvious to show it.(adsbygoogle = window.adsbygoogle || []).push({});

I'm working with random variables. So the [itex]L^2[/itex] norm of X is [itex]E(X^2)^{1/2}[/itex], where E is the expected value. Thus, we want to show: if [itex]E((X_n-X)^2)\to0[/itex], then [itex]E(X_n^2)\to E(X^2)[/itex].

From [itex]E((X_n^2-X)^2)\to0[/itex], we get

[tex]E(X_n^2)\to2E(X_nX)-E(X^2).[/tex]

I think it should be true that [itex]2E(X_nX)\to2E(X^2)[/itex], which would prove the result, but I'm not sure how to prove that.

Any help?

Or a reference? Is the result in a book?

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# Does L^2 Convergance Imply Convergance of L^2 norms?

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