Gauss–Kuzmin distribution

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
Probability mass function
PDF of the Gauss Kuzmin Distribution
Cumulative distribution function
CDF of the Gauss Kuzmin Distribution
Parameters (none)
Support
PMF
CDF
Mean
Median
Mode
Variance
Skewness (not defined)
Excess kurtosis (not defined)
Entropy 3.432527514776...[1][2][3]

Gauss–Kuzmin theorem

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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

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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

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References

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  1. ^ Blachman, N. (1984). "The continued fraction as an information source (Corresp.)". IEEE Transactions on Information Theory. 30 (4): 671–674. doi:10.1109/TIT.1984.1056924.
  2. ^ Kornerup, Peter; Matula, David W. (July 1995). "LCF: A Lexicographic Binary Representation of the Rationals". J.UCS the Journal of Universal Computer Science. Vol. 1. pp. 484–503. CiteSeerX 10.1.1.108.5117. doi:10.1007/978-3-642-80350-5_41. ISBN 978-3-642-80352-9. {{cite book}}: |journal= ignored (help)
  3. ^ Vepstas, L. (2008), Entropy of Continued Fractions (Gauss-Kuzmin Entropy) (PDF)
  4. ^ Weisstein, Eric W. "Gauss–Kuzmin Distribution". MathWorld.
  5. ^ Gauss, Johann Carl Friedrich. Werke Sammlung. Vol. 10/1. pp. 552–556.
  6. ^ Kuzmin, R. O. (1928). "On a problem of Gauss". Dokl. Akad. Nauk 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 dépendant les quotients complets et incomplets d'une fraction continue". Bulletin de la Société Mathématique de France. 57: 178–194. doi:10.24033/bsmf.1150. 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 (5): 507–528. doi:10.4064/aa-24-5-507-528.
  10. ^ Babenko, K. I. (1978). "On a problem of Gauss". Soviet Math. Dokl. 19: 136–140.