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

: $p(k) = - \log_2 \left( 1 - \frac\right)~.$

Gauss–Kuzmin theorem

Let

:$x = \cfrac$

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

:$\lim_ \mathbb \left\ = - \log_2\left(1 - \frac\right)~.$

Equivalently, let

:$x_n = \cfrac~;$

then

:$\Delta_n(s) = \mathbb \left\ - \log_2(1+s)$

tends to zero as ''n'' tends to infinity.

Rate of convergence

In 1928, Kuzmin gave the bound

:$, \Delta_n(s), \leq C \exp(-\alpha \sqrt)~.$

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

:$\Psi(s) = \lim_ \frac$

exists for every ''s'' in , 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

References

{{DEFAULTSORT:Gauss-Kuzmin distribution
Continued fractions
Discrete distributions