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

TL;DR

This paper investigates the convergence rate of a noncommutative central limit theorem and analyzes the return sequence of a specific dynamical system in the upper half-plane, revealing new probabilistic and dynamical properties.

Contribution

It provides a Berry-Esseen type estimate for the monotone CLT and characterizes the return sequence for singular measures in parabolic iteration.

Findings

Established a convergence speed estimate for the monotone CLT.

Proved the dynamical system preserves infinite measure with a regularly varying return sequence.

Identified the return sequence's index as 1/2 for singular measures.

Abstract

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.