Gauss–Kuzmin distribution

Parameters (none) $k \in \{1,2,\ldots\}$ $-\log_2\left[ 1-\frac{1}{(k+1)^2}\right]$ $1 - \log_2\left(\frac{k+2}{k+1}\right)$ $+\infty$ $2\,$ $1\,$ $+\infty$ (not defined) (not defined) 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]