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

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, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Reciprocal Triangular U-quadratic uniform Wigner semicircle

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson S_U Landau Laplace Asymmetric Laplace Linnik 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 distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled 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 discrete 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