Gauss–Kuzmin distribution

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]

References

↑ 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.
↑ 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.
↑ Vepstas, L. (2008), Entropy of Continued Fractions (Gauss-Kuzmin Entropy) (PDF)
↑ Weisstein, Eric W. "Gauss–Kuzmin Distribution". MathWorld.
↑ Gauss, C.F. Werke Sammlung. 10/1. pp. 552–556.
↑ Kuzmin, R.O. (1928). "On a problem of Gauss". DAN SSSR: 375–380.
↑ Kuzmin, R.O. (1932). "On a problem of Gauss". Atti del Congresso Internazionale dei Matematici, Bologna. 6: 83–89.
↑ 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.
↑ Wirsing, E. (1974). "On the theorem of Gauss–Kusmin–Lévy and a Frobenius-type theorem for function spaces". Acta Arithmetica. 24: 507–528.
↑ Babenko, K.I. (1978). "On a problem of Gauss". Soviet Math. Dokl. 19: 136–140.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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Gauss–Kuzmin distribution

Gauss–Kuzmin theorem

Rate of convergence

See also

References