Yet another proof of the Nualart-Peccati criterion
Résumé
In 2005, Nualart and Peccati showed that, surprisingly, the convergence in distribution of a normalized sequence of multiple Wiener-Itô integrals towards a standard Gaussian law is equivalent to convergence of just the fourth moment to 3. Very recently, this result was extended to a sequence of multiple Wigner integrals, in the context of free Brownian motion. The goal of the present paper is to offer an elementary, unifying proof of these two latter results. The only advanced, needed tool is the product formula for multiple integrals. Unless this formula, the rest of the proof only relies on soft combinatorial arguments.
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