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Weak convergence of the sum of dependent variables, question

  1. Dec 13, 2011 #1
    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.
     
  2. jcsd
  3. Dec 15, 2011 #2
    How about simply Yn = -Xn?
     
  4. Dec 15, 2011 #3

    mathman

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    This is not a counterexample Xn -> X, Yn -> Y (=-X) Xn + Yn -> 0 (= X + Y).
     
  5. Dec 16, 2011 #4
    With weak convergence you could set Y iid to -X instead of Y=-X (so that X+Y <> 0 if X is non-trivial).
     
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