  In this paper we prove the existence of asymptotic moments, and an estimate
on the tails of the limiting distribution, for a specific class of almost
periodic functions. Then we introduce the hyperbolic circle problem, proving an
estimate on the asymptotic variance of the remainder that improves a result of
Chamizo. Applying the results of the first part we prove the existence of
limiting distribution and asymptotic moments for three functions that are
integrated versions of the remainder, and were considered originally (with due
adaptations to our settings) by Wolfe, Phillips and Rudnick, and Hill and
Parnovski.
