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Thread: 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 $\displaystyle Y-X_n$ (which is non-negative), and to use the fact that $\displaystyle \limsup_n (a-u_n)=a-\liminf_n u_n$.

    Notice as well that $\displaystyle E[X_n]$ makes sense since the positive part of $\displaystyle X$ is integrable ($\displaystyle X_+\leq |Y|$), so that you can write $\displaystyle E[Y-X_n]=E[Y]-E[X_n]$ (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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