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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.

Fact that {Xn+Yn}-->{X+Y} weakly, can be shown using characteristic functions and Levy's theorem.

Question:

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

I appreciate any help in advance.

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# Weak convergence of the sum of dependent variables, question

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