Statistics Seminar: Joe Neeman (University of Texas Austin) -- Gaussian vectors, half-spaces, and convexity

Abstract:

Let A be a subset of R^n and let B be a half-space with the same Gaussian
measure as A. For a pair of correlated Gaussian vectors X and Y, Pr(X \in
A, Y \in A) is smaller than Pr(X \in B, Y \in B); this was originally
proved by Borell, who also showed various other extremal properties of
half-spaces. For example, the exit time of an Ornstein-Uhlenbeck process
from A is stochastically dominated by its exit time from B.

We will discuss these (and other) inequalities using a kind of modified convexity.