Talk by Todd Kemp (UCSD)

Title: Chaos and the Fourth Moment

Abstract: In 2005, Nualart and Peccati proved a very surprising central limit theorem for multiple Wiener-Itô integrals: if X_k is a (variance-normalized) sequence of nth Wiener integrals (for fixed n ≥ 2), then X_k --> N(0,1) in distribution if and only if E(X_k^4) --> 3. That is: in a fixed order of Wiener chaos, convergence to a normal distribution is equivalent to the a priori dramatically weaker convergence of the fourth moment alone.

In this talk, I will discuss extensions of this theorem to the world of random matrices. Matrix chaos is built from Wiener integrals with respect to matrix-valued Brownian motion. In the limit as matrix size goes to infinity, a coherent theory emerges describing a chaos of limiting eigenvalue distributions. In this context, Nualart and Peccati's theorem has an exact analogue, but the condition for convergence is E(X_k^4) --> 2 (the fourth moment of the semicircle law). The tools of Malliavin calculus are used to give quantitative estimates on the distance to the limit distribution.

This is joint work with Ivan Nourdin, Giovanni Peccati, and Roland Speicher.