7
$\begingroup$

I know there are different versions of the central limit theorem and consequently there are different proofs of it. The one I am most familiar with is in the context of a sequence of identically distributed random variables, and the proof is based on an integral transform (eg. characteristic function, moment generating function), followed by first order approximations to obtain a function to which the inverse transform can be applied.

I am interested to know if there are any flaws in this approach - I have been told informally that it is not completely rigorous - but why ?

Improve this question
$\endgroup$
3
  • 5
    $\begingroup$ This post by Terence Tao might be of interest. $\endgroup$
    – user10525
    Commented May 18, 2012 at 8:16
  • 1
    $\begingroup$ I think a complete proof can be given using characteristic functions. It depends on the argument you use to know whether or not the proof is complete. Also as you point there are different versions of the CLT. They differ on the conditions put on the population distribution of the random variables you average. Lyaponov's is fairly general but Lindeberg-Feller is more general. $\endgroup$
    – Michael R. Chernick
    Commented May 18, 2012 at 12:01
  • $\begingroup$ When I was a graduate student I had tp learn a proof of the Lindeberg-Fellwr Version of the Central limit theorem for my Probability Theory Class we used Jacque Neveu's book The Calculus of Probability and the Second Volume Feller's Theory of Probability THeory book. I think the formal proof was in Neveu'a book. $\endgroup$
    – Michael R. Chernick
    Commented May 20, 2012 at 2:06

1 Answer 1

Reset to default
2
$\begingroup$

As I recall in this version the random variables are independent with finite variances but the variance need not all be the same. The CLT result holds under a somewhat complicated condition called the Lindeberg condition and the traditional proofs use transform methods.
But the proof we learned was probabilistic. It involved splitting the sum into two pieces. One piece converged to N(0,1) in distribution and the other converge to 0 in probability. This technique was used because it was much easier to show the first sum satisfied the CLT. But the fact that the second sum was negligible was harder. The following link gives an interesting paper by Larry Goldstein that give a probabilistic proof of the Linderberg Feller Theorem that is very similar or the same. It also may be of interest to the OP because it includes some history on the CLT. http://bcf.usc.edu/~larry/papers/pdf/lin.pdf

Improve this answer
$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.