Talk:Gauss–Kuzmin distribution

Recherches Arithmétiques in Gauss-Kuzmin distribution

 * Copied from User talk:Linas:

You added Recherches Arithmétiques as a reference to Gauss-Kuzmin distribution a couple of years back. I wondered where you found this. Unfortunately, I don't have access to the original, and I couldn't find it in skimming the Gauss' Collected Works (Werke Sammlung). I'd love to confirm the reference &mdash; for example, giving a page number &mdash; but I can't seem to, so I'm curious how you did. Thanks, Calbaer 15:50, 12 May 2007 (UTC)


 * Linas will hopefully be able to give a more complete reply. All I can tell you is that Recherches Arithmétiques is a translation in French of Disquisitiones Arithmeticae. -- Jitse Niesen (talk) 03:31, 13 May 2007 (UTC)


 * I'm pretty sure that observation was not made in Disquisitiones Arithmeticae. According to Kuzmin's original paper, it was made in a letter to Laplace written after Disquisitiones Arithmeticae was published.  According to Knuth (TAOCP v.2 ed.2 p.347), prior to that it was merely in his notebook.  Kuzmin says that the letter is in volume 10 of the complete works, and, after poking around a fair bit, I found an entry in Gauss' "journal with explanations" on pp. 552-556 of Band 10, Abt 1 (of the edition linked above).  The reference should be modified accordingly, unless there's also a mention of it in Disquisitiones Arithmeticae as well (which, again I doubt). Calbaer 05:40, 13 May 2007 (UTC)


 * Although I am fortunate in having access to a very good library (UT Austin), I would not be able to verify the reference i.e. come up with a page number, without going there, and spending half a day crawling through the thing. Which is not something I feel like doing just now. I no longer recall what prompted me to put the Recherches Arithmétiques reference in there, I must have been hasty. We should, however, move this conversation to the talk page of that article. linas 17:53, 13 May 2007 (UTC)


 * Yes, the half-day at the library is what I wound up doing, though I came up somewhat empty before continuing the search online at night. (I wanted to have a solid reference for an academic paper.)  Like I say, though, all references I can find seem to indicate it's elsewhere, and I'm afraid that the "Recherches Arithmétiques" reference has by now shown up in an academic paper, so I'll correct the link, and, if you find a reference in Disquisitiones Arithmeticae, you can correct my correction. Calbaer 22:10, 13 May 2007 (UTC)

Entropy
The page currently states that the entropy is 3.43... but surely this cannot be right; or, perhaps, some other definition of entropy is being used? A quick numerical calculation of


 * $$H=-\sum_{k=0}^\infty p_k \log p_k$$

where
 * $$p_k=\log_2\left[1-\frac{1}{(1+k)^2}\right]$$

shows that
 * $$H=2.379246...$$

although the sum is rather slow to converge ... Now one has that
 * $$\frac{2.379246...}{\log(2)}= 3.4325...$$

but one does not normally define entropy with this extra factor of 2... oh, hmm. Appearently information-entropists, according to our wikipedia article do define their entropy with log_2. OK, whatever ... linas (talk) 20:33, 1 June 2008 (UTC)

Convergence speed.
The article actually states that the speed of convergence of continuous fractions' digits to the G.-K. distribution is exponential. In fact, it is true, but only for one of two possible interpretations of the problem. You can either take a Lebesgue-typical point, and then consider the distribution of its digits -- and then there's no chance for the exponential convergence speed (as, taking n digits, you can't be closer than 1/n to the limiting distribution). Or you can take the Lebesgue measure, and ask for the distribution of n-th digit of a Lebesgue-random point x. And in this second way of asking things you indeed have the exponential convergence. But maybe it's better to be precised? Because, the initial way how G.-K. statistic appears, is indeed the behaviour of individual number "digits"? Regards, Burivykh (talk) 21:16, 27 April 2009 (UTC)
 * Yes, this should be clarified. linas (talk) 20:56, 11 April 2012 (UTC)

Starting value must be irrational
The article assumes a random variable uniformly distributed in (0, 1), but it appears to me that we must take (0, 1) \ Q, since the continued-fraction expansions of the rationals terminate. Since the probability of randomly hitting upon a rational number is zero, this is a purely theoretical issue, but I think we should steer clear from division by zero even in theory. --Lambiam 20:05, 24 May 2014 (UTC)
 * as you wrote, the probability of a rational is zero, so we may follow the usual convention that random variables are defined modulo events of probability zero. Sasha (talk) 19:37, 26 May 2014 (UTC)
 * It is not clear to me what it means for a random variable to be "defined modulo events of probability zero". Take the r.v. for the uniform distribution on [0, 1). Any elementary event realizing the r.v. has probability zero. So this r.v. is defined "modulo all these elementary events"??? Can I find more about this convention in Wikipedia? Would this solve the issue what the domain of the distribution is? --Lambiam 23:46, 26 May 2014 (UTC)
 * I am no longer very active on Wikipedia, so I do not have much to say about Wiki conventions. In the textbooks on probability theory which I have studied, a random variable is defined as a class of equivalence of measurable functions from the probability space, where two functions are called equivalent if they coincide on a measurable set of full probability. If we start from a function and call it a random variable, we actually mean its equivalence class (and this is the usual convention, similar to that in L p spaces et cet.), and by the same convention a function which is not defined at all on a set of zero probability is considered a random variable (since the equivalence class does not depend on the choice of the values on a set of zero probability). This is what I had in mind. Sasha (talk) 02:12, 29 May 2014 (UTC)

More general result
In the article in MathWorld we find a more general result: iteration of the map $T$(ξ) = {1/ξ}, where {_} denotes the fractional part of a real number, gives in the limit the distribution function log2(1+$x$). The discrete version for the coefficients of the continued fraction presented now in the article is easily derived from this. Should this more general result be given in the article?

It is elementary to verify that this continuous distribution (defined on (0, 1) \ Q) is a fixpoint of the operator corresponding to $T$, but this may be "original research". --Lambiam 20:20, 24 May 2014 (UTC)