Convergence of Wigner integrals to the tetilla law

TL;DR

This paper establishes a precise criterion for the convergence of sequences of multiple Wigner integrals to the tetilla law, a specific distribution arising from free probability, based on moments convergence.

Contribution

It provides a new moment-based characterization for convergence to the tetilla law in the context of free probability, extending previous limit theorems.

Findings

Convergence occurs if and only if the fourth and sixth moments converge.

The tetilla law is characterized by its relation to free semicircular variables.

The result parallels similar theorems in classical probability.

Abstract

If x and y are two free semicircular random variables in a non-commutative probability space (A,E) and have variance one, we call the law of 2^{-1/2}(xy+yx) the tetilla law (and we denote it by T), because of the similarity between the form of its density and the shape of the tetilla cheese from Galicia. In this paper, we prove that a unit-variance sequence {F_n} of multiple Wigner integrals converges in distribution to T if and only if E[F_n^4]--> E[T^4] and E[F_n^6]--> E[T^6]. This result should be compared with limit theorems of the same flavor, recently obtained by Kemp, Nourdin, Peccati & Speicher and Nourdin & Peccati.