Proof of Central Limit Theorem

The central limit theorem is one of the most important theorems in statistics. It approximates the distribution of the meanof any sample of size n from any distribution by the distribution

Now write

so we have

 
 

 
 

 
 

 
 

 
 

 
 

 
 

 
 

Now expand so

since

 
 

 
 

 

,

 

Taking the Fourier transform,

 
 

This is of the form

where and.

But this is a Fourier transform of a Gaussian function, so

Therefore,

 
 

 
 

But and, so