Talk:Khinchin's theorem

Misnomer?

I'm really not convinced that the elementary result in this article is commonly referred to as "Khinchin's theorem". I ran a search on Google Scholar, and the most common reference to "Khinchin's theorem" is in connection with Diophantine approximation (specifically, to the probabilistic distribution of integers in "almost all" regular continued fraction expansions of real numbers). That topic is covered in Wikipedia under Khinchin's constant and also under Levy's constant.

It also appears that Khinchin's name appears most often in connection with other mathematicians when used to refer to a theorem (e.g., Wiener-Khinchin theorem, Bochner-Khinchin theorem, Birkhoff-Khinchin theorem). I'll try looking with alternative transliterations of his name, but right now it appears to me that this article is factually incorrect. DavidCBryant 13:58, 1 June 2007 (UTC)

I ran another Google Scholar search, on "Khintchine theorem", with the same result. "Wiener-Khintchine theorem" shows up a lot, along with the Diophantine approximation stuff, but I didn't find any references to the existence of an inverse mapping (which seems so obvious it hardly deserves a special name). I also checked my old copy of Feller's book, where I find Khintchine mentioned in connection with a certain asymptotic equality (in a problem set), with the law of the iterated logarithm (Bernoulli trials), and with the law of large numbers. That's it.

I think I'm going to propose this article for deletion based on this research. DavidCBryant 15:45, 1 June 2007 (UTC)

I think Kinchin (and his 1928/1929 theorem) should be mentioned in Law of large numbers (LLN) article. Giftlite 17:55, 4 June 2007 (UTC)

David, you are a.s. right. I've corrected the spelling errors and such, but have grave doubts that this has much to do with Khinchin, besides the fact that he worked on probability theory, among, of course, many other subjects. On the other hand, the elementary fact itself might be notable, only a probabilist can tell. Arcfrk 00:09, 7 June 2007 (UTC)

I prodded this article, and only later realized it has been prodded (unsuccessfully) before. Anyways, I support David here, and I will AFD this article if needed.(Igny 04:04, 1 August 2007 (UTC))

Old variant

I put it here for now. Khinchin's theorem (named after Aleksandr Khinchin) is a mathematical theorem used in probability theory.

Formulation

Let F be any continuous cumulative distribution function and let U be a random variable ~ U[0,1], meaning that U is uniformly distributed over the unit interval. Then the cumulative distribution function of the random variable X = F ⁻¹(U) is F.

Proof

If we denote by F_X the cumulative distribution function of the random variable X = F ⁻¹(U) then:

$F_{X}(x)=P({X<x})=P({F^{-1}(U)<x})=P({U<F(x)})=F(x),$

meaning that the cumulative distribution function of X is indeed F.