A free central-limit theorem for dynamical systems

TL;DR

This paper extends the free central-limit theorem to dynamical systems of operators, introducing free mixing coefficients and providing bounds, with connections to classical probability and random matrix theory.

Contribution

It generalizes the free CLT to dynamical systems using free mixing coefficients and establishes Berry-Essen bounds, broadening its applicability.

Findings

Free CLT holds for certain dynamical systems with mixing conditions.

Berry-Essen bounds are established for these systems.

Connections to classical probability and random matrix theory are demonstrated.

Abstract

The free central-limit theorem, a fundamental theorem in free probability, states that empirical averages of freely independent random variables are asymptotically semi-circular. We extend this theorem to general dynamical systems of operators that we define using a free random variable $X$ coupled with a group of *-automorphims describing the evolution of $X$. We introduce free mixing coefficients that measure how far a dynamical system is from being freely independent. Under conditions on those coefficients, we prove that the free central-limit theorem also holds for these processes and provide Berry-Essen bounds. We generalize this to triangular arrays and U-statistics. Finally we draw connections with classical probability and random matrix theory with a series of examples.