Gauss–Kuzmin distribution

From Wikipedia, the free encyclopedia
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

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

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

References

  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. 
 
Discrete univariate with infinite support
This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.