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Milds # 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...

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). The distribution is named after Carl Friedrich Gauss, who derived it around 1800, and Rodion Kuzmin, who gave a bound on the rate of convergence in 1929. 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 improved it to

$|\Delta _{n}(s)|\leq C\,0.7^{n}~.$ Later, Eduard Wirsing showed 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.