Proof of central limit theorem

[skwey]

Hi I want to prove this using momentgenerating functions. I would like to do this without going into 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

2.
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:

M'(t/n)*t/M(t/n)

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?

 

[skwey]

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.