Proof of Central Limit Theorem

[chingkui]

I have been reading some books about the proof of the Central Limit Theorem, all of them use the uniqueness of moment generating function. But since I have not yet seen a proof of the uniqueness properties, is there any proof that does not use this result? Thanks.

 

[Krusty]

It can be done in terms of Characteristic functions also. A brief proof is given on the wikipedia site for Central Limit Theorem. Uniqueness of a characteristic function holds because it is just the Fourier transform of the corresponding density function, up to a multiplicative constant

 

[ssd]

I have been reading some books about the proof of the Central Limit Theorem, all of them use the uniqueness of moment generating function. But since I have not yet seen a proof of the uniqueness properties, is there any proof that does not use this result? Thanks.

Moment generating functions are not unique in general. Proof of CLT uses characteristic function and CF's are unique.

 

[chingkui]

I am not familiar with the characteristic function, is there a place I can quickly read about its uniqueness? Is characteristic function a necessary step in existing CLT proofs? Thanks.

 

[ZioX]

1) http://tt.lamf.uwindsor.ca/65-540/540Files/11.pdf
2) http://tt.lamf.uwindsor.ca/65-540/540Files/13.pdf

You need a lot of background to prove this result, which is why it's often skipped in undergraduate courses.

 

[ljxaa]

ch.f is not the only tool for proving CLT, however in proper setting it is quick and convinient; as far as i know, stein's method another approach