Berry-Esseen type estimate and return sequence for parabolic iteration in the upper half-plane

Two different aspects of parabolic iteration in the complex upper half-plane are considered here. First, from a noncommutative probability perspective, a Berry-Esseen type estimate for the convergence speed of the monotone central limit theorem is proved. Secondly, if the underlying measure in this central limit process is singular to the Lebesgue measure on the real line, then the iteration is shown to be an infinite-measure preserving dynamical system that has a regularly varying return sequence of index 1/2.