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. 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

Gauss–Kuzmin theorem

Let
be the continued fraction expansion of a random number x uniformly distributed in. Then
Equivalently, let
then
tends to zero as n tends to infinity.

Rate of convergence

In 1928, Kuzmin gave the bound
In 1929, Paul Lévy improved it to
Later, Eduard Wirsing showed that, for λ=0.30366..., the limit
exists for every s in , and the function Ψ is analytic and satisfies Ψ=Ψ=0. Further bounds were proved by K.I.Babenko.