Free Quantitative Fourth Moment Theorems on Wigner Space

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.