Central limit problem

#1
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

TS Contributor
#2
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
 
#3
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