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Math Help - Limiting distribution - iid gamma random variables

  1. #1
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    Limiting distribution - iid gamma random variables

    here is another question i stuck. again if you help me and explain steps, i will be pleased. thanks again

    **
    Suppose that
    X1, X 2, , X n be i.i.d. random variables with Gamma (α,n) pdf with mean μ= nα and variance σ^2=αn^2. Let Ynα in probability. Find the limiting distribution of

    sqrt(Y
    n)* [sqrt(
    sampleXn / nα - 1 ) ]


    **
    Suppose that X1, X2,, Xn are i.i.d. Uniform(0,1). Denote U n=sqrt(n)*(sample Xn − 1/2) , n 1.

    Find the moment generating function of Un.
    Find the limiting distribution Un using first part
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  2. #2
    MHF Contributor matheagle's Avatar
    Joined
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    I get....

    If X_i\sim\Gamma(\alpha,n)

    then \biggl({\bar X_n\over \alpha n}-1\biggr)\sqrt{\alpha n} converges to a standard normal
    and of course by slutsky I can replace that alpha with Y's
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