Weak convergence of the sum of dependent variables, question

[vovchik]

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.
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.

 

[bpet]

How about simply Yn = -Xn?

 

[mathman]

How about simply Yn = -Xn?

This is not a counterexample Xn -> X, Yn -> Y (=-X) Xn + Yn -> 0 (= X + Y).

 

[bpet]

This is not a counterexample Xn -> X, Yn -> Y (=-X) Xn + Yn -> 0 (= X + Y).

With weak convergence you could set Y iid to -X instead of Y=-X (so that X+Y <> 0 if X is non-trivial).