Central Limit Theorems for a Stationary Semicircular Sequence in Free Probability

TL;DR

This paper establishes a central limit theorem for functionals of stationary semicircular sequences with long-range dependence in free probability, extending classical Gaussian results to a non-commutative setting.

Contribution

It introduces a free probability analogue of the classical CLT for non-linear functionals of stationary sequences with long-range dependence.

Findings

Proves a CLT for functionals of stationary semicircular sequences

Extends classical Gaussian CLT results to free probability setting

Handles long-range dependence in the sequence

Abstract

In this paper, we focus on studying central limit theorems for functionals of some specific stationary random processes. In classical probability theory, it is well-known that for non-linear functionals of stationary Gaussian sequences, we can get a central-limit result via Hermite polynomials and the diagram formula for cumulants. In this paper, the main result is an analogous central limit theorem, in a free probability setting, for real-valued functionals of a stationary semicircular sequence with long-range dependence, namely the correlation function of the underlying time series tends to zero as the lag goes to infinity.