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
Letbe the continued fraction expansion of a 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 boundIn 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.