Gauss–Kuzmin distribution

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Gauss–Kuzmin
Parameters (none)
Support k \in \{1,2,\ldots\}
pmf -\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)
Ex. kurtosis (not defined)
Entropy 3.432527514776...[1][2][3]

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).[4] The distribution is named after Carl Friedrich Gauss, who derived it around 1800,[5] and Rodion Kuzmin, who gave a bound on the rate of convergence in 1929.[6][7] It is given by the probability mass function

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

Gauss–Kuzmin theorem[edit]

Let

 x = \frac{1}{k_1 + \frac{1}{k_2 + \cdots}}

be the continued fraction expansion of a random 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 = \frac{1}{k_{n+1} + \frac{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[edit]

In 1928, Kuzmin gave the bound

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

In 1929, Paul Lévy[8] improved it to

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

Later, Eduard Wirsing showed[9] 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.[10]

See also[edit]

References[edit]

  1. ^ Blachman, N. (1984). "The continued fraction as an information source (Corresp.)". IEEE Transactions onInformation Theory 30 (4): 671–674. doi:10.1109/TIT.1984.1056924. 
  2. ^ Kornerup, P.; Matula, D. (July 1995). "LCF: A lexicographic binary representation of the rationals". Journal of Universal Computer Science 1: pp. 484–503. 
  3. ^ Vepstas, L. (2008), Entropy of Continued Fractions (Gauss-Kuzmin Entropy) 
  4. ^ Weisstein, Eric W., "Gauss–Kuzmin Distribution", MathWorld.
  5. ^ Gauss, C.F. Werke Sammlung 10/1. pp. 552–556. 
  6. ^ Kuzmin, R.O. (1928). "On a problem of Gauss". DAN SSSR: 375–380. 
  7. ^ Kuzmin, R.O. (1932). "On a problem of Gauss". Atti del Congresso Internazionale dei Matematici, Bologna 6: pp. 83–89. 
  8. ^ Lévy, P. (1929). "Sur les lois de probabilité dont dépendent les quotients complets et incomplets d'une fraction continue". Bulletin de la Société Mathématique de France 57: pp. 178–194. JFM 55.0916.02. 
  9. ^ Wirsing, E. (1974). "On the theorem of Gauss–Kusmin–Lévy and a Frobenius-type theorem for function spaces". Acta Arithmetica 24: pp. 507–528. 
  10. ^ Babenko, K.I. (1978). "On a problem of Gauss". Soviet Math. Dokl. 19: pp. 136–140.