PDA

View Full Version : Central limit problem

Gekko
08-23-2010, 09:42 PM
Hi, I've been struggling with the below and have no idea how to tackle it. I've looked at the mgf but this doesn't seem to work out.
Intuitively I imagine if we can obtain the mgf of the normal then the proof is complete? Ie exp(t^2/2) for mean 0 and var 1

Any help greatly appreciated. Stumped!!

Z=(-1/sqrt(n)) * sum from k=1 to n of [1+log(1-Fk(Xk))]

where Fk is a cumulative distribution function which is continious and strictly increasing. Xk is independent random variable

Show that as n->infinity, Z converges to a normal distribution with mean 0 and var 1

BGM
08-23-2010, 11:01 PM
Note the following facts (You may prove them if you like):

1. If F_k is the cumulative distribution function of
X_k , then F_k(X_k) \sim Uniform(0, 1)

If U \sim Uniform(0, 1) , then

2. 1 - U \sim Uniform(0, 1)

3. E[\ln U ] = -1

4. Var[\ln U] = 1

Then I guess you have the enough information for you to apply the
Central Limit Theorem

Gekko
08-24-2010, 10:11 AM
BGM

Big thank you. Your tips were enough for me to solve this and to prove each step

Thanks for taking the time to reply. It really helped me

Dason
08-24-2010, 10:30 AM
Note that 1) is only true for a continuous r.v.