  We establish two precise asymptotic results on the Birkhoff sums for
dynamical systems. These results are parallel to that on the arithmetic sums of
independent and identically distributed random variables previously obtained by
Hsu and Robbins, Erd\H{o}s, Heyde. We apply our results to the Gauss map and
obtain new precise asymptotics in the theorem of L\'evy on the regular
continued fraction expansion of irrational numbers in $(0,1)$.
