Normal approximation for iterated inner functions

TL;DR

This paper establishes a Berry--Eséen theorem for linear combinations of iterates of inner functions, providing a new proof of a central limit theorem for such functions using elementary transfer and martingale theory.

Contribution

It introduces a Berry--Eséen theorem for inner function iterates and offers a simplified proof of an existing central limit theorem in this context.

Findings

Proved a Berry--Eséen theorem for linear combinations of inner function iterates.

Provided a simple proof of Nicolau and Soler i Gibert's central limit theorem for inner functions.

Utilized elementary transfer arguments and classical martingale results.

Abstract

A Berry--Ess\'{e}en theorem for linear combinations of iterates of an inner function is obtained. Our proof, which is based an elementary transfer argument and classical results in martingale theory, also leads to a simple proof of Nicolau and Soler i Gibert's central limit theorem for inner functions.