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


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