We consider the diophantine approximation problem
We present how metric theory of continued fraction expansion together
with probabilistic results for weakly dependent random variables can
be used in order to investigate the asymptotic behaviour of the above
sequences of random variables. Especially, we present strong
invariance principles and invariance principles in distribution for
the sequence and a central limit theorem for the sequence
(which solves a conjecture of LeVeque) obtained by that method.