Gauss–Kuzmin distribution

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Gauss–Kuzmin
Probability mass function
PDF of the Gauss Kuzmin Distribution
Cumulative distribution function
CDF of the Gauss Kuzmin Distribution
Parameters (none)
Support k{1,2,}
PMF log2[11(k+1)2]
CDF 1log2(k+2k+1)
Mean +
Median 2
Mode 1
Variance +
Skewness (not defined)
Excess 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)=log2(11(1+k)2).

Gauss–Kuzmin theorem

Let

x=1k1+1k2+

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

limn{kn=k}=log2(11(k+1)2).

Equivalently, let

xn=1kn+1+1kn+2+;

then

Δn(s)={xns}log2(1+s)

tends to zero as n tends to infinity.

Rate of convergence

In 1928, Kuzmin gave the bound

|Δn(s)|Cexp(αn).

In 1929, Paul Lévy[8] improved it to

|Δn(s)|C0.7n.

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

Ψ(s)=limnΔn(s)(λ)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 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.