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Math Help - Urgent homework on expectation

  1. #1
    UMD
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    Urgent homework on expectation

    Can anyone help me in proving the following question

    Let {Xn} be a sequence of random variables satisfying Xn <= Y a.s for some Y with E(|Y|) < infinity. then show that

    E( Lim n approaches infinity Sup Xn) > = lim n approaches infinity Sup E (Xn)



    Thanking inadvance
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  2. #2
    MHF Contributor

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    I think that the all you have to do is to apply Fatou's lemma to Y-X_n (which is non-negative), and to use the fact that \limsup_n (a-u_n)=a-\liminf_n u_n.

    Notice as well that E[X_n] makes sense since the positive part of X is integrable ( X_+\leq |Y|), so that you can write E[Y-X_n]=E[Y]-E[X_n] (where infinite values are possible).
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  3. #3
    UMD
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    thanks

    Thanks Alot. I will look into it. I have a few more questions which i will post today. i hope u will help me in those questions too.again very thanks

    Regards

    Uzair


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