# Free quantitative fourth moment theorems on Wigner space

## TL;DR

This paper establishes a quantitative Fourth Moment Theorem for Wigner integrals, providing new tools for analyzing convergence to the semicircular distribution in free probability, with applications to free fractional Brownian motion.

## Contribution

It generalizes previous results by proving a quantitative theorem for Wigner integrals of any order with symmetric kernels, using free stochastic analysis and a new biproduct formula.

## Key findings

- Quantitative bounds for convergence to semicircular law
- Extension of Fourth Moment Theorem to higher-order Wigner integrals
- Berry-Esseen bounds for free fractional Brownian motion

## Abstract

We prove a quantitative Fourth Moment Theorem for Wigner integrals of any order with symmetric kernels, generalizing an earlier result from Kemp et al. (2012). The proof relies on free stochastic analysis and uses a new biproduct formula for bi-integrals. A consequence of our main result is a Nualart-Ortiz-Latorre type characterization of convergence in law to the semicircular distribution for Wigner integrals. As an application, we provide Berry-Esseen type bounds in the context of the free Breuer-Major theorem for the free fractional Brownian motion.