The central limit theorem for monotone convolution with applications to free Levy processes and infinite ergodic theory

TL;DR

This paper extends the central limit theorem to monotone convolution, using free harmonic analysis to characterize norming constants, and applies these results to construct new ergodic measure-preserving transformations on the real line.

Contribution

It provides a complete characterization of norming constants in the monotonic CLT and introduces new conservative ergodic transformations based on these findings.

Findings

Complete characterization of norming constants in the monotonic CLT

Construction of new conservative ergodic measure-preserving transformations

Generalized Boole transformation shown to be conservative under certain conditions

Abstract

In this paper free harmonic analysis tools are used to study parabolic iteration in the complex upper half-plane. The main result here is a complete characterization for the norming constants in the monotonic central limit theorem. This allows us to construct a new class of conservative and ergodic measure-preserving transformations on the real line with Lebesgue measure. Among all, we mention that the generalized Boole transformation with infinitely many poles is shown to be conservative, as long as its residues are summable.