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Math Help - Weak convergence of the sum of dependent variables, question

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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.
    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.
    Last edited by vovchik; December 13th 2011 at 03:39 PM.
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    Re: Weak convergence of the sum of dependent variables, question

    Hello,

    You can just consider a symmetric rv : Xn such that Xn has the same distribution as -Xn.

    So we can consider Xn and Yn=-Xn, which is not independent of Xn. It is true to say that Xn converges to a rv X and Yn converges to a rv Y that follows the same distribution as X.
    But Xn+Yn=0 doesn't converge to X+Y.

    For example P(X_n=1)=P(X_n=-1)=\frac 12
    It converges weakly to X, such that P(X=1)=P(X=-1)=\frac 12.
    Same for Y_n=-X_n and Y.
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    Re: Weak convergence of the sum of dependent variables, question

    Thank you!
    Last edited by vovchik; December 13th 2011 at 08:51 PM.
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