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Math Help - Limiting Distribution

  1. #1
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    Limiting Distribution

    Let X_1,X_2,...,X_n be i.i.d. random variables from N(1,1) distribution. Find the limiting distribution of the ran. var.

    W_n = sqrt(n)*[(X_1 + X_2 + ... + X_n - n)] / [((X_1 - 1)^2) + ... + ((X_n - 1)^2)].

    We have recently seen this limiting distributions but haven't solved enough problems about it and i don't know which theorem i should apply for a problem(though i guess central limit theorem should work for this problem). Please help me with this. Thanks!
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  2. #2
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    Start by finding the moment generating function of Wn, ie E(e^{W_n t}), once you have found it, take it's limit as n approaches infinity.
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  3. #3
    MHF Contributor matheagle's Avatar
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    It looks like your trying to get a T, but there are 3 problems

    \sum_{i=1}^nX_i\sim N(n,n)

    thus {\sum_{i=1}^nX_i-n\over\sqrt{n}}\sim N(0,1)

    and \sum_{i=1}^n(X_i-1)^2\sim \chi_n^2

    NOW a t with a degrees of freedom is a {N(0,1)\over\sqrt{\chi^2_a/a}}\sim t_a

    So, you're missing another sqrt of n, the sqrt on the denominator
    BUT moreover, these two sums don't seem to be independent.
    Maybe we can use that \bar X is independent of S^2
    -------------------------------------------------------------------------------------------
    Got it, I was trying to find it's EXACT distribution, but limiting is easy.
    Use two facts, which I already said...

    {\sum_{i=1}^nX_i-n\over\sqrt{n}}\sim N(0,1)

    \sum_{i=1}^n(X_i-1)^2\sim\chi_n^2

    Thus by the SLLN {\sum_{i=1}^n(X_i-1)^2\over n}\to E(\chi_1^2)=1

    take the ratio and you are done, the limiting result is a N(0,1)

    {{\sum_{i=1}^nX_i-n\over\sqrt{n}}\over {\sum_{i=1}^n(X_i-1)^2\over n}}\to {N(0,1)\over 1}

    This is the CLT plus Slutsky.
    Actually the numerator is a N(0,1), so it's not really a CLT.
    Last edited by matheagle; December 1st 2009 at 08:48 PM.
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