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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 ?

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    $\begingroup$ This post by Terence Tao might be of interest. $\endgroup$ – user10525 May 18 '12 at 8:16
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    $\begingroup$ 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 May 18 '12 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 May 20 '12 at 2:06
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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 trandform methods. But the proof we learned was probabilistic. It involved spliting 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 probabilidtiv 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

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