November 5, 2009:  Todd Kemp (Assistant Professor, University of California, San Diego and Visiting Assistant Professor, MIT, 2009 - 2010)

Chaos and the Fourth Moment

Abstract:  The Wiener Chaos is a natural orthogonal decomposition of the L^2 space of a Brownian motion, naturally associated to stochastic integration theory; the orders of chaos are given by the range of multiple Wiener-Ito integrals.

In 2006, Nualart and collaborators proved a remarkable central limit theorem in the context of the chaos.  If $X_k$ is a sequence of $n$th Wiener-Ito integrals (in the $n$th chaos), then necessary and sufficient conditions that $X_k$ converge weakly to a normal law are that its (second and) fourth moments converge -- all other moments are controlled by these.

In this lecture, I will discuss recent joint work with Roland Speicher in which we prove an analogous theorem for the empirical eigenvalue laws of high-dimensional random matrices.