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Gauss–Kuzmin distribution

Probability distribution in number theory

Summary

Probability distribution in number theory

name =Gauss–Kuzmin| type =mass| pdf_image = [[File:GaussKuzminPDF.png|PDF of the Gauss Kuzmin Distribution]] support =k \in {1,2,\ldots}| pdf =-\log_2\left[ 1-\frac{1}{(k+1)^2}\right]| cdf =1 - \log_2\left(\frac{k+2}{k+1}\right)| mean =+\infty| median =2,| mode =1,| variance =+\infty| skewness =(not defined)| kurtosis =(not defined)| entropy =3.432527514776...| mgf =| char =|}}

In mathematics, the Gauss–Kuzmin distribution is a discrete probability distribution that arises as the limit probability distribution of the coefficients in the continued fraction expansion of a random variable uniformly distributed in (0, 1). The distribution is named after Carl Friedrich Gauss, who derived it around 1800, and Rodion Kuzmin, who gave a bound on the rate of convergence in 1929.{{cite journal |first=R. O. |last=Kuzmin |title=On a problem of Gauss

: p(k) = - \log_2 \left( 1 - \frac{1}{(k+1)^2}\right)~.

Gauss–Kuzmin theorem

Let

: x = \cfrac{1}{k_1 + \cfrac{1}{k_2 + \cdots}}

be the continued fraction expansion of a number x uniformly distributed in (0, 1). Then

: \lim_{n \to \infty} \mathbb{P} \left{ k_n = k \right} = - \log_2\left(1 - \frac{1}{(k+1)^2}\right)~.

Equivalently, let

: x_n = \cfrac{1}{k_{n+1} + \cfrac{1}{k_{n+2} + \cdots}}~;

then

: \Delta_n(s) = \mathbb{P} \left{ x_n \leq s \right} - \log_2(1+s)

tends to zero as n tends to infinity.

Rate of convergence

In 1928, Kuzmin gave the bound

: |\Delta_n(s)| \leq C \exp(-\alpha \sqrt{n})~.

In 1929, Paul Lévy improved it to

: |\Delta_n(s)| \leq C , 0.7^n~.

Later, Eduard Wirsing showed that, for λ = 0.30366... (the Gauss–Kuzmin–Wirsing constant), the limit

: \Psi(s) = \lim_{n \to \infty} \frac{\Delta_n(s)}{(-\lambda)^n}

exists for every s in [0, 1], and the function Ψ(s) is analytic and satisfies Ψ(0) = Ψ(1) = 0. Further bounds were proved by K. I. Babenko.

References

References

  1. (1984). "The continued fraction as an information source (Corresp.)". IEEE Transactions on Information Theory.
  2. (July 1995). "J.UCS the Journal of Universal Computer Science". [[Journal of Universal Computer Science]].
  3. Vepstas, L.. (2008). "Entropy of Continued Fractions (Gauss-Kuzmin Entropy)".
  4. "Gauss–Kuzmin Distribution".
  5. Gauss, Johann Carl Friedrich. "Werke Sammlung".
  6. Kuzmin, R. O.. (1928). "On a problem of Gauss". Dokl. Akad. Nauk SSSR.
  7. Lévy, P.. (1929). "Sur les lois de probabilité dont dépendant les quotients complets et incomplets d'une fraction continue". [[Bulletin de la Société Mathématique de France]].
  8. Wirsing, E.. (1974). "On the theorem of Gauss–Kusmin–Lévy and a Frobenius-type theorem for function spaces". Acta Arithmetica.
  9. Babenko, K. I.. (1978). "On a problem of Gauss". Soviet Math. Dokl..
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