Gauss–Kuzmin distribution

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.

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]

• Khinchin's constant
• Lévy's constant