# Freeness characterizations on free chaos spaces

## TL;DR

This paper provides new characterizations of freeness in free chaos spaces using contraction operators, covariance, and free Malliavin gradients, aiding in limit theorems and asymptotic analysis.

## Contribution

It introduces three novel characterizations of freeness in free chaos spaces, enhancing understanding and analysis of free stochastic processes.

## Key findings

- Three characterizations of freeness: contraction operators, covariance, free Malliavin gradients
- Applications to limit theorems and asymptotic properties
- Framework for analyzing convergence in free probability

## Abstract

This paper deals with characterizing the freeness and asymptotic freeness of free multiple integrals with respect to a free Brownian motion or a free Poisson process. We obtain three characterizations of freeness, in terms of contraction operators, covariance conditions, and free Malliavin gradients. We show how these characterizations can be used in order to obtain limit theorems, transfer principles, and asymptotic properties of converging sequences.