Berry–Esseen Type Estimate and Return Sequence for Parabolic Iteration in the Upper Half-Plane

We prove a Berry–Esseen type theorem for the convergence rate in the monotone central limit theorem. When the underlying measure is singular relative to Lebesgue measure, this central limit process is viewed as an infinite measure-preserving dynamical system and we prove that it has a regularly varying return sequence of index $1/2$.