# Central limit problem

#### Gekko

##### New Member
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
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

##### New Member
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