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

: <math> p(k) = - \log_2 \left( 1 - \frac{1}{(k+1)^2}\right)~.</math>

Gauss&ndash;Kuzmin theorem

Let

:<math> x = \cfrac{1}{k_1 + \cfrac{1}{k_2 + \cdots </math>

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

:<math> \lim_{n \to \infty} \mathbb{P} \left\{ k_n = k \right\} = - \log_2\left(1 - \frac{1}{(k+1)^2}\right)~.</math>

Equivalently, let

:<math> x_n = \cfrac{1}{k_{n+1} + \cfrac{1}{k_{n+2} + \cdots~; </math>

then

:<math> \Delta_n(s) = \mathbb{P} \left\{ x_n \leq s \right\} - \log_2(1+s) </math>

tends to zero as n tends to infinity.

Rate of convergence

In 1928, Kuzmin gave the bound

:<math> |\Delta_n(s)| \leq C \exp(-\alpha \sqrt{n})~. </math>

In 1929, Paul Lévy improved it to

:<math> |\Delta_n(s)| \leq C \, 0.7^n~. </math>

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

:<math> \Psi(s) = \lim_{n \to \infty} \frac{\Delta_n(s)}{(-\lambda)^n} </math>

exists for every s in [0,&nbsp;1], and the function Ψ(s) is analytic and satisfies Ψ(0)&nbsp;=&nbsp;Ψ(1)&nbsp;=&nbsp;0. Further bounds were proved by K. I. Babenko.

See also

  • Khinchin's constant
  • Lévy's constant

References