Proof of Central Limit Theorem

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SUMMARY

The discussion centers on the proof of the Central Limit Theorem (CLT) and the use of characteristic functions as an alternative to moment generating functions. Participants confirm that characteristic functions are unique and can be utilized in CLT proofs, unlike moment generating functions, which are not unique in general. Resources such as Wikipedia and specific academic PDFs are recommended for further reading on the uniqueness of characteristic functions and their role in CLT proofs. Additionally, Stein's method is mentioned as another viable approach for proving the CLT.

PREREQUISITES
  • Understanding of Central Limit Theorem (CLT)
  • Familiarity with characteristic functions and their properties
  • Knowledge of moment generating functions
  • Basic concepts of Fourier transforms
NEXT STEPS
  • Read about the uniqueness of characteristic functions in probability theory
  • Study Stein's method for proving the Central Limit Theorem
  • Explore the relationship between moment generating functions and characteristic functions
  • Review the proof of the Central Limit Theorem using characteristic functions
USEFUL FOR

Mathematicians, statisticians, and students studying probability theory, particularly those interested in advanced proofs of the Central Limit Theorem and the properties of characteristic functions.

chingkui
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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.
 
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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
 
chingkui said:
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.
 
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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.
 
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.
 
Last edited by a moderator:
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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