## The Annals of Probability

### Wigner chaos and the fourth moment

#### Abstract

We prove that a normalized sequence of multiple Wigner integrals (in a fixed order of free Wigner chaos) converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2, the fourth moment of the semicircular law. This extends to the free probabilistic, setting some recent results by Nualart and Peccati on characterizations of central limit theorems in a fixed order of Gaussian Wiener chaos. Our proof is combinatorial, analyzing the relevant noncrossing partitions that control the moments of the integrals. We can also use these techniques to distinguish the first order of chaos from all others in terms of distributions; we then use tools from the free Malliavin calculus to give quantitative bounds on a distance between different orders of chaos. When applied to highly symmetric kernels, our results yield a new transfer principle, connecting central limit theorems in free Wigner chaos to those in Gaussian Wiener chaos. We use this to prove a new free version of an important classical theorem, the Breuer–Major theorem.

#### Article information

Source
Ann. Probab., Volume 40, Number 4 (2012), 1577-1635.

Dates
First available in Project Euclid: 4 July 2012

https://projecteuclid.org/euclid.aop/1341401144

Digital Object Identifier
doi:10.1214/11-AOP657

Mathematical Reviews number (MathSciNet)
MR2978133

Zentralblatt MATH identifier
1277.46033

#### Citation

Kemp, Todd; Nourdin, Ivan; Peccati, Giovanni; Speicher, Roland. Wigner chaos and the fourth moment. Ann. Probab. 40 (2012), no. 4, 1577--1635. doi:10.1214/11-AOP657. https://projecteuclid.org/euclid.aop/1341401144

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