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Proof of Central Limit Theorem

  1. Feb 26, 2007 #1
    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.
  2. jcsd
  3. Feb 26, 2007 #2
    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
  4. Feb 27, 2007 #3


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    Moment generating functions are not unique in general. Proof of CLT uses characteristic function and CF's are unique.
    Last edited: Feb 27, 2007
  5. Feb 27, 2007 #4
    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.
  6. Feb 28, 2007 #5
    1) http://tt.lamf.uwindsor.ca/65-540/540Files/11.pdf [Broken]
    2) http://tt.lamf.uwindsor.ca/65-540/540Files/13.pdf [Broken]

    You need a lot of background to prove this result, which is why it's often skipped in undergraduate courses.
    Last edited by a moderator: May 2, 2017
  7. Mar 9, 2007 #6
    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:cool:
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