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Here is the question:

Let $\xi_n $ be a sequence of random variables on a probability space $(\Omega,\mathcal{F},P) $ such that $E \xi_n^2 \le c $ for some constant $c$. Assume that $\xi_n \to \xi $ almost surely as $n \to \infty$. Prove that $E \xi $ is finite and $E \xi_n \to E \xi $.

I guess the condition that $ E \xi_n^2 \le c$ is really important here. But I don't know how to use it correctly.

Did
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andy90
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  • Which theorems do you know which would ensure that $E\xi_n\to E\xi$? – Did Oct 28 '14 at 20:37
  • I know the dominated convergence theorem and monotonic convergence theorem. But it seems that none of it can be used here. – andy90 Oct 28 '14 at 20:41
  • OK, then you might want to try this: http://en.wikipedia.org/wiki/Uniform_integrability – Did Oct 28 '14 at 20:44
  • Not sure I understand your comment. See http://en.wikipedia.org/wiki/Uniform_integrability#Relation_to_convergence_of_random_variables – Did Oct 28 '14 at 20:56

1 Answers1

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Thanks to the suggestion of Did, I'm able to answer this question now. The answer is as follows.

$E \xi_n^2 \le c$ means that $\{ \xi_n \} $ is uniformly bounded in $L^2(\Omega,\mathcal{F},P)$, which suggests that $\{ \xi_n \}$ is uniformly integrable.

$E\xi_n^2 \le c$ implies that $E |\xi_n| \le \sqrt{c}$, which further implies $|\xi_n | < \infty$ almost surely. Since $\xi_n \to \xi$ almost surely, we are able to say that $|\xi| < \infty$ almost surely.

Now, we can use Vitali convergence theorem, which implies that $E \xi$ is finite and $E |\xi_n - \xi| \to 0 $ as $n \to \infty $.

andy90
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