Gauss–Kuzmin distribution

Parameters (none) (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

Gauss–Kuzmin theorem

Let

be the continued fraction expansion of a random number x uniformly distributed in (0, 1). 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[8] improved it to

Later, Eduard Wirsing showed[9] that, for λ=0.30366... (the Gauss-Kuzmin-Wirsing constant), the limit

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]