Convergence in law in the second Wiener/Wigner chaos

TL;DR

This paper identifies conditions under which convergence in law in the second Wiener chaos can be established by checking finitely many cumulants, extending Nualart and Peccati's result to broader classes and free probability.

Contribution

It introduces a large subset of limiting laws where finite cumulant convergence guarantees law convergence, extending known results to non-Gaussian and free probability contexts.

Findings

Finite cumulant convergence implies law convergence in a large subset of second Wiener chaos limits.

Analogous results hold for free Brownian motion and Wigner integrals.

Extension of Nualart and Peccati's fourth cumulant criterion to broader settings.

Abstract

Let L be the class of limiting laws associated with sequences in the second Wiener chaos. We exhibit a large subset L_0 of L satisfying that, for any F_infinity in L_0, the convergence of only a finite number of cumulants suffices to imply the convergence in law of any sequence in the second Wiener chaos to F_infinity. This result is in the spirit of the seminal paper by Nualart and Peccati, in which the authors discovered the surprising fact that convergence in law for sequences of multiple Wiener-It\^o integrals to the Gaussian is equivalent to convergence of just the fourth cumulant. Also, we offer analogues of this result in the case of free Brownian motion and double Wigner integrals, in the context of free probability.