In honor of Herman Rubin

Speaker(s)

Steven P. Lalley (University of Chicago)
Richard Bradley (Indiana University)
Frederi Viens (Purdue University)

Description

This session honors the career and achievements of Herman Rubin, who has been a professor in the Purdue departments of Statistics and Mathematics since 1967, and is a prodigy in twentieth century statistics. The speakers in the session are all prominent probabilists, which appropriately honors Herman's extensive contributions to probability, only one of Herman's countless topics of interest. The relatively narrow focus of the session should not obscure the breadth of Herman's numerous insightful work in mathematics and statistics, ranging from topology, logic, and set theory to density estimation, asymptotics, Bayesian decision theory, inequalities, total positivity and random number generation.

Schedule

Fri, June 22 - Location: STEW 214

Time	Speaker	Title

10:30-10:55AM	Steven P. Lalley	Spatial Extent of a Critical Branching Random Walk
	Abstract: In a branching random walk (BRW), particles alternately reproduce, as in a Galton-Watson process, and then move in space, as in an ordinary random walk. When the offspring distribution has mean 1, the BRW is said to be critical, and when the step distribution of the associated random walk has mean 0 the BRW is said to be driftless. In a critical BRW the population of particles will eventually die out, with probability 1, so not all points of the ambient space will be visited. We will address the following question: If a critical, driftless BRW is initiated by a single particle at the origin, what is the probability that some particle finds its way out of the ball of radius R centered at the origin before the process dies out?
11:00-11:25AM	Richard Bradley	On the behavior of the covariance matrices in a multivariate central limit theorem under some mixing conditions
	Abstract: In a paper that appeared in 2010, C. Tone proved a multivariate central limit theorem for some strictly stationary random fields of random vectors satisfying certain mixing conditions. The ''normalization of a given ''partial sum (or ''block sum) involved matrix multiplication by a ''standard negative one-half power of its covariance matrix (a symmetric, positive definite matrix), and the limiting multivariate normal distribution had the identity matrix as its covariance matrix. The mixing assumptions in Tone's result implicitly imposed an upper bound on the ratios of the largest to the smallest eigenvalues in the covariance matrices of the partial sums. It turns out that in Tone's result, for the entire collection of the covariance matrices of the partial sums, there is essentially no other restriction on the relative magnitudes of the eigenvalues or on the (orthogonal)directions of the corresponding eigenvectors. For simplicity, the example discussed in this talk to illustrate this point, will involve just random sequences, not the broader context of random fields. The sequence that we discuss will be a stationary Gaussian sequence of random vectors. For the case of strictly stationary sequences of (univariate) random variables, Tone's result boils down to a central limit theorem of M. Peligrad that appeared in 1996, and the construction described above is partly adapted from a corresponding univariate (Gaussian) construction of the author that appeared in 1999.
11:30 - 11:55AM	Frederi Viens	Density, tail, and expectation comparison inequalities on Wiener space
	Abstract: A recent device introduced by Ivan Nourdin and Giovanni Peccati allows the use of a combination of tools from stochastic analysis on Wiener space and Stein's equations to estimate distances between distributions of random variables that are differentiable in the sense of Malliavin, and classical target distributions such normal and Gamma laws. We will discuss some new developments in this direction, by which a randomized measure of spread can be used to extend classical theorems about expected functionals of Gaussian fields to non-Gaussian ones, with possible applications to polymers in non-Gaussian environments and to extensions of the Sherrington-Kirkpatrick model. Some of this methodology appears to require no reference to Stein's equations. Some basic background on the Malliavin calculus will be given, in an attempt to make the presentation largely self-contained. This is joint work in progress with I. Nourdin and G. Peccati. Attendees at this presentation are encouraged to also attend the presentations of Juan-Jose Viquez and Richard Eden (in the Session on Probability: Algorithms, Stochastic Analysis, and Applications part I) where the connection between the Nourdin-Peccati device and Stein's equations will be emphasized.