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

${\displaystyle p(k)=-\log _{2}\left(1-{\frac {1}{(1+k)^{2}}}\right)~.}$

Gauss–Kuzmin

Probability mass function
Cumulative distribution function
Parameters	(none)
Support	${\displaystyle k\in \{1,2,\ldots \}}$
PMF	${\displaystyle -\log _{2}\left[1-{\frac {1}{(k+1)^{2}}}\right]}$
CDF	${\displaystyle 1-\log _{2}\left({\frac {k+2}{k+1}}\right)}$
Mean	${\displaystyle +\infty }$
Median	${\displaystyle 2\,}$
Mode	${\displaystyle 1\,}$
Variance	${\displaystyle +\infty }$
Skewness	(not defined)
Excess kurtosis	(not defined)
Entropy	3.432527514776...[1][2][3]

Gauss–Kuzmin theorem

Let

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

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

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

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]

See also

• Khinchin's constant
• Lévy's constant