Convergence in law in the second Wiener/Wigner chaos
نویسنده
چکیده
Let L be the class of limiting laws associated with sequences in the second Wiener chaos. We exhibit a large subset L0 ⊂ L satisfying that, for any F∞ ∈ L0, 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∞. This result is in the spirit of the seminal paper [12], in which Nualart and Peccati discovered the surprising fact that convergence in law for sequences of multiple Wiener-Itô 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.
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