##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability and Statistics Seminar

## Todd Kemp

#### UCSD, 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.

Host: Bruce Driver

### November 5, 2009

### 9:00 AM

### AP&M 6402

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