Gauss–Kuzmin distribution

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Gauss–Kuzmin
Parameters (none)
Support
pmf
CDF
Mean
Median
Mode
Variance
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

Gauss–Kuzmin theorem[edit]

Let

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

Equivalently, let

then

tends to zero as n tends to infinity.

Rate of convergence[edit]

In 1928, Kuzmin gave the bound

In 1929, Paul Lévy[8] improved it to

Later, Eduard Wirsing showed[9] that, for λ=0.30366... (the Gauss-Kuzmin-Wirsing constant), the limit

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: 484–503. doi:10.1007/978-3-642-80350-5_41. 
  3. ^ Vepstas, L. (2008), Entropy of Continued Fractions (Gauss-Kuzmin Entropy) (PDF) 
  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: 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: 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: 507–528. 
  10. ^ Babenko, K.I. (1978). "On a problem of Gauss". Soviet Math. Dokl. 19: 136–140.