Korman Center, Room 245, 15 S. 33rd Street, Philadelphia, PA 19104
Colloquium: The Dirichlet Curve of a Probability in R^d
Wednesday, October 16, 2013
3:00 PM-4:00 PM
Gérard Letac, Université de Toulouse
Abstract: A random Dirichlet distribution $P_t$ on $R^d$, of intensity $t>0$ and governed by the probability $\alpha(dx)$ on $R^d,$ is such that for any partition $(A_0,\ldots,A_n)$ the distribution of $(P_t(A_0),\ldots,P_t(A_n))$ is Dirichlet with parameters $(t\alpha(A_0),\ldots,t\alpha(A_n).$ Therefore the expectation $X_t=\int xP_t(dx)$ is a random variable and the study of its distribution $\mu(t\alpha)$ is challenging. The map $t\mapsto mu(t\alpha)$ is called the Dirichlet curve of $\alpha.$ We prove that its limit when $t$ goes to infinity is Cauchy or Dirac. In the Dirac case, we show that $t\mapsto\int f(x) mu(t\alpha)$ is decreasing for any positive convex function (some would say that $t\mapsto mu(t\alpha)$ is decreasing in the Strassen order). This is joint work with Mauro Piccioni.
We will recall what a Dirichlet distribution and a Cauchy distribution in $R^d$ are and we will explain through the Sethuraman construction that $P_t$ is a purely atomic distribution whose atoms are dense in the support of $\alpha.$ The proofs of our results use unexpected facts about the ordinary beta distributions, and the lecture can be followed by students.