Central limit theorems in linear dynamics

TL;DR

This paper investigates conditions under which sequences derived from bounded linear operators on Banach spaces converge in distribution to Gaussian variables, extending the understanding of central limit phenomena in linear dynamics.

Contribution

It introduces new criteria for the existence of measures ensuring Gaussian convergence for sums of functions composed with linear operators.

Findings

Established conditions for Gaussian convergence in linear dynamics

Identified classes of functions for which the CLT holds

Extended classical CLT results to operator-induced sequences

Abstract

Given a bounded operator $T$ on a Banach space $X$, we study the existence of a probability measure $\mu$ on $X$ such that, for many functions $f:X\to\mathbb K$, the sequence $(f+\dots+f\circ T^{n-1})/\sqrt n$ converges in distribution to a Gaussian random variable.