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Math Help - limit of random variables, expectation

  1. #1
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    limit of random variables, expectation

    Hallo!

    Can anybody please help me?


    Is the following equality true?
    \lim\limits_{h \to 0}\mathbb{E}[X_h]=\mathbb{E}[X]

    X is a random variable X>0
    X_h is a sequence of random variables, 1>h>0, X_h>0

    with
    \mathbb{E}[X_h] \leq \mathbb{E}[X]
    p\lim\limits_{h \to 0}X_h=X, where p denotes convergence in probability


    Now, I think it follows with Fatou's Lemma for non-negative random variables, that
    \liminf\limits_{h \to 0}\mathbb{E}[X_h]\geq \mathbb{E}[\liminf\limits_{h \to 0}X_h].

    Can I now say, that
    \lim\limits_{h \to 0}\mathbb{E}[X_h]=\mathbb{E}[X]
    is true?

    Thanks in advance!
    Last edited by Juju; March 25th 2011 at 10:24 AM.
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  2. #2
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    Whoops, didn't read one of the conditions. I will think about this more.
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  3. #3
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    OK. Thanks!
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  4. #4
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    The issue I have with your logic is the invocation of Fatou's Lemma. You should need almost sure convergence for that. Everything else goes through fine I think (if you have Fatou's Lemma then you can get \limsup \mathbb E X_h \le \mathbb E X \le \liminf \mathbb E X_h).
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  5. #5
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    Okay, Fatou's Lemma looks fine. Apparently all you need is convergence in probability for that. Sorry for any confusion my ignorance may have caused
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  6. #6
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    So, you think under the given assumptions
     \lim\limits_{h \to 0}EX_h=EX is correct?

    But do you have any idea how to show it?
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  7. #7
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    Okay, this question is starting to make me quite nervous. The first inequality I posted should come from the fact that \mathbb E X_h \le \mathbb E X and the second inequality is the version of Fatou's Lemma I was able to find that concerned convergence in probability - that is, \mathbb E X \le \liminf \mathbb E X_h provided we have convergence in probability, i.e. you can get rid of the liminf.
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  8. #8
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    This question is making me very nervous.
    But thank you so much for helping me,

    Julia
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