  We study the distribution of the sequence of continued fraction convergents
$p_n/q_n$ to a random irrational number in the group
$\mathrm{SL}(2,\mathbb{Z}_m)$, and in particular the distribution of $p_n
\pmod{m}$ and $q_n \pmod{m}$ with a fixed modulus $m$. Improving the strong law
of large numbers due to Sz\"usz, Moeckel, Jager and Liardet, we establish the
central limit theorem and the law of the iterated logarithm, as well as the
weak and the almost sure invariance principles. As an application, we find the
limit distribution of the maximum and the minimum of the Birkhoff sum for the
irrational rotation with the indicator of an interval as test function. We also
compute the normalizing constant in a classical limit law for the same Birkhoff
sum due to Kesten, and dispel a misconception about its dependence on the test
interval.
