From: hrubin@b.stat.purdue.edu (Herman Rubin) Newsgroups: sci.math Subject: Re: Central Limit Theorem and Arc Length Date: 12 Dec 1998 18:24:31 -0500 In article <74rger$qfe$1@nnrp1.dejanews.com>, wrote: >In article <74pj1h$31c$1@news.duke.edu>, > "Kamran Sajadi" wrote: >> 1. Can anyone direct me towards a book that has a good proof of the Central >> Limit Theorem? Here's what I hope to find: something of the graduate >> level, NOT one from an undergraduate probability course textbook. This is >> because I want a proof that doesn't simply use the convergence of the >> moment-generating function, but something more substantial. ............... > You can find a proof of the Central Limit Theorem in >"any" book on probability (the one on my shelf would be >Chung "A Course in Probability Theory"). I'm not certain >what your criteria mean. This book, as most others, use a characteristic function proof. This is quite unfortunate, as this still disguises what is going on, although I do not think that there is any proof which REALLY shows what is happening. The Lindeberg proof, which only uses undergraduate mathematics, is 100% rigorous, and has full generality. Usually some cute devices are used which requires more mathematics, but these are unnecessary. Even the Lindeberg-Feller theorem can be proved this way. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 ============================================================================== From: "Kamran Sajadi" Newsgroups: sci.math Subject: Re: Central Limit Theorem and Arc Length Date: Sat, 12 Dec 1998 22:46:44 -0500 >This book, as most others, use a characteristic function >proof. This is quite unfortunate, as this still disguises >what is going on, although I do not think that there is >any proof which REALLY shows what is happening. Yeah, the convergence of the characteristic function proves to be pretty useful. It is the critical step of a proof that involves the Fourier Inversion formula and the shows that the actual Probability distribution converges to the normal distribution. This proof is really interesting because it really does show what's happening. Using Fourier analysis, you begin by seeing that the characteristic function E(exp(itX)) of random variable X is equal to Int (-infinity to infinity) [exp(itx) * x^2 * f(x) dx] where f is the density function, and this is just the Fourier transform of the density function. You bounce stuff around, prove that the characteristic function converges to exp (-(t^2)/2), use the Fourier Inversion Formula, and you get the fourier transform of the Gaussian with sum junk that you can easily manipulate into the normal distribution (details omitted). I may have screwed this description up, i'm not looking at the proof. Anyway, those details are pretty straightforward, I just needed to find a book which shows the convergence of the characteristic function, which now have -- thanks to the suggestions from this newsgroup.