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 (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

$p(k)=-{\log }_2(1-\frac{1}{(k+1{)}^2}).$

Gauss–Kuzmin theorem

Let

$x=\frac{1}{k_1+\frac{1}{k_2+\cdots }}$

be the continued fraction expansion of a random number x uniformly distributed in (0, 1). Then

$\underset{n\to \mathrm{\infty }}{\lim }\mathbb{\mathbb{P}}\{k_n=k\}=-{\log }_2(1-\frac{1}{(k+1{)}^2}).$

Equivalently, let

$x_n=\frac{1}{k_{n+1}+\frac{1}{k_{n+2}+\cdots }};$

then

${\mathrm{\Delta }}_n(s)=\mathbb{\mathbb{P}}\{x_n\le s\}-{\log }_2(1+s)$

tends to zero as n tends to infinity.

Rate of convergence

In 1928, Kuzmin gave the bound

$|{\mathrm{\Delta }}_n(s)|\le C\exp (-\alpha \sqrt{n}).$

In 1929, Paul Lévy improved it to

$|{\mathrm{\Delta }}_n(s)|\le C0.7^n.$

Later, Eduard Wirsing showed that, for λ = 0.30366... (the Gauss–Kuzmin–Wirsing constant), the limit

$\mathrm{\Psi }(s)=\underset{n\to \mathrm{\infty }}{\lim }\frac{{\mathrm{\Delta }}_n(s)}{(-\lambda {)}^n}$

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.

See also

• Khinchin's constant
• Lévy's constant