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Proof of central limit theorem

  1. Dec 4, 2011 #1
    Hi I want to prove this using momentgenerating functions. I would like to do this without going in to the standard normal distribution, just the normal distribution.

    I would like to show that the momentgenerating function of
    (x1+x2+x3....xn)/n--->e^(ut+sigma^2t/2) as n-->infinity.

    x1, x2, x3,xn =independent variables with mean u and variance sigma^2
    e^(ut+sigma^2t/2)=momentgenerating function of a normal distribution

    1.calculating the moment generating function
    This I get to be

    [M(t/n)]^n where M(t) is the momentgenerating function of the variable x1 or x2 or xn. [M(t/n)]^n is the momentgenerating function of (x1+x2+x3..xn)/n

    Finding the limit as n->infinity.

    I take the natural logarithm and get n*ln[M(t/n)]=ln[M(t/n)]/(1/n)
    as n->infinity we get 0/0 since M(0)=1 and ln(1)=0

    I then use l'Hopital to get:


    when n goes to infinity this goes to ut since M'(0)=u, but it should be ut+sigma^2/2

    Does anyone see why I do not get the last part, what have I forgotten?
    Last edited: Dec 4, 2011
  2. jcsd
  3. Dec 4, 2011 #2
    The l'Hopital part:

    ln[M(t/n)]' / (1/n) '
    =[M'(t/n)*-t/n^2 / M(t/n)] / [-1/n^2]= M'(t/n)*t/M(t/n). As n-> infinity this becomes ut.
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