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
 formula_1
Gauss–Kuzmin theorem.
Let
formula_2
be the continued fraction expansion of a random number "x" uniformly distributed in (0, 1). Then
formula_3
Equivalently, let
formula_4
then
formula_5
tends to zero as "n" tends to infinity.
Rate of convergence.
In 1928, Kuzmin gave the bound
formula_6
In 1929, Paul Lévy improved it to
formula_7
Later, Eduard Wirsing showed that, for "λ" = 0.30366... (the Gauss–Kuzmin–Wirsing constant), the limit
formula_8
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.
References.
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