Precise asymptotics on the Birkhoff sums for dynamical systems

Abstract

We establish two precise asymptotic results on the Birkhoff sums for dynamical systems. These results are parallel to that on the partial sums of independent and identically distributed random variables previously obtained by Hsu and Robbins, Erdős, Heyde. We apply our results to the Gauss map and obtain new precise asymptotics in the theorem of Lévy on the regular continued fraction expansion of irrational numbers in (0, 1).