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December 13th, 2011, 02:15 PM   #1
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Weak convergence of the sum of dependent variables question

Hi guys,

Problem: Let {Xn},{Yn} - real-valued random variables.
{Xn}->{X} - weakly; {Yn}->{Y} weakly.
Assume that Xn and Yn - independent for all n and that X and Y - are independent. Show that {Xn+Yn}->{X+Y} weakly.
This can be shown using Levy's theorem and characteristic functions.

Question:
If independence does not hold, can you construct a counterexample?

I appreciate any help in advance.
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