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


 * $$ p(k) = - \log_2 \left( 1 - \frac{1}{(1+k)^2}\right)~.$$

Gauss–Kuzmin theorem
Let


 * $$ x = \cfrac{1}{k_1 + \cfrac{1}{k_2 + \cdots}} $$

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


 * $$ \lim_{n \to \infty} \mathbb{P} \left\{ k_n = k \right\} = - \log_2\left(1 - \frac{1}{(k+1)^2}\right)~.$$

Equivalently, let


 * $$ x_n = \cfrac{1}{k_{n+1} + \cfrac{1}{k_{n+2} + \cdots}}~; $$

then


 * $$ \Delta_n(s) = \mathbb{P} \left\{ x_n \leq s \right\} - \log_2(1+s) $$

tends to zero as n tends to infinity.

Rate of convergence
In 1928, Kuzmin gave the bound


 * $$ |\Delta_n(s)| \leq C \exp(-\alpha \sqrt{n})~. $$

In 1929, Paul Lévy improved it to


 * $$ |\Delta_n(s)| \leq C \, 0.7^n~. $$

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


 * $$ \Psi(s) = \lim_{n \to \infty} \frac{\Delta_n(s)}{(-\lambda)^n} $$

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.