  In this paper we prove the following renewal-type limit theorem. Given an
irrational $\alpha$ in (0,1) and R>0, let $q_{n_R}$ be the first denominator of
the convergents of $\alpha$ which exceeds R. The main result in the paper is
that the ratio $q_{n_R}/R$ has a limiting distribution as R tends to infinity.
The existence of the limiting distribution uses mixing of a special flow over
the natural extension of the Gauss map.
