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