Free Quantitative Fourth Moment Theorems on Wigner Space

Abstract

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.

A quantitative fourth moment theorem in free probability theory

“…for an ad hoc distance d C 2 which is in fact weaker than the Wassertein distance W 2 (see Lemma 3.16). Whether a similar fourth moment bound holds for chaoses of higher orders, as in the commutative setting (see Nualart and Peccati [NP05] and Nourdin and Peccati [NP09]), is a question which has first been investigated by Bourguin and Campese in [BC17]. They provided the following bound…”

“…By restricting themselves to the case of fully symmetric kernels, Bourguin and Campese prove the following quantitative bound. Proposition 3.14 (Corollary 3.8 of [BC17]). Let n ≥ 2, and let F be a self-adjoint element in the homogeneous Wigner chaos H n , which can be written F = I n (f ) for a fully symmetric f ∈ L 2 (R n + ), and such that τ (F 2 ) = 1.…”

“…The proof of Theorem 1.1 uses the notions of free Stein kernel and free Stein discrepancy introduced by Fathi and Nelson in [FN17] (see Section 2), and the free Malliavin calculus as defined by Biane and Speicher in [BS98] (see Section 3). One can consult [BC17] and the reference therein to get a larger overview about the classical and free results related to the "fourth moment theorem".…”

Freeness characterizations on free chaos spaces

“…A crucial tool in the analysis of Wigner integrals is the product formula (1), and a biproduct formula for bi-integrals was recently obtained in [5], which will be a crucial tool in the sequel. It makes use of a new type of contraction, referred to in [5] as bicontractions, defined as follows. Let…”

“…In [5], a quantitative version of Theorem 1.3 is derived, using free stochastic analysis as well as a new biproduct formula for bi-integrals.…”