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

${\displaystyle p(k)=-\log _{2}\left(1-{\frac {1}{(1+k)^{2}}}\right)~.}$
Parameters (none) ${\displaystyle k\in \{1,2,\ldots \}}$ ${\displaystyle -\log _{2}\left[1-{\frac {1}{(k+1)^{2}}}\right]}$ ${\displaystyle 1-\log _{2}\left({\frac {k+2}{k+1}}\right)}$ ${\displaystyle +\infty }$ ${\displaystyle 2\,}$ ${\displaystyle 1\,}$ ${\displaystyle +\infty }$ (not defined) (not defined) 3.432527514776...[1][2][3]

Gauss–Kuzmin theorem

Let

${\displaystyle x={\frac {1}{k_{1}+{\frac {1}{k_{2}+\cdots }}}}}$

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

${\displaystyle \lim _{n\to \infty }\mathbb {P} \left\{k_{n}=k\right\}=-\log _{2}\left(1-{\frac {1}{(k+1)^{2}}}\right)~.}$

Equivalently, let

${\displaystyle x_{n}={\frac {1}{k_{n+1}+{\frac {1}{k_{n+2}+\cdots }}}}~;}$

then

${\displaystyle \Delta _{n}(s)=\mathbb {P} \left\{x_{n}\leq s\right\}-\log _{2}(1+s)}$

tends to zero as n tends to infinity.

Rate of convergence

In 1928, Kuzmin gave the bound

${\displaystyle |\Delta _{n}(s)|\leq C\exp(-\alpha {\sqrt {n}})~.}$

In 1929, Paul Lévy[8] improved it to

${\displaystyle |\Delta _{n}(s)|\leq C\,0.7^{n}~.}$

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

${\displaystyle \Psi (s)=\lim _{n\to \infty }{\frac {\Delta _{n}(s)}{(-\lambda )^{n}}}}$

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]

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. Journal of Universal Computer Science. 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.
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. 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. 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.