Proof of Central Limit Theorem
The cenral 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
