Georgia Institute of Technology College of Sciences

We review various classifications of probability distributions based on their tail heaviness. Using a characterization of medium-tailed distributions we propose a test for testing the null hypothesis of medium-tail vs long- or short-tailed distributions. Some operating characteristics of the proposed test are discussed.

Excursion sets of stationary random fields have attracted much attention in recent years.They have been applied to modeling complex geometrical structures in tomography, astro-physics and hydrodynamics. Given a random field and a specified level, it is natural to studygeometrical functionals of excursion sets considered in some bounded observation window.Main examples of such functionals are the volume, the surface area and the Euler charac-teristics. Starting from the classical Rice formula (1945), many results concerning calculationof moments of these geometrical functionals have been proven. There are much less resultsconcerning the asymptotic behavior (as the window size grows to infinity), as random variablesconsidered here depend non-smoothly on the realizations of the random field. In the talk wediscuss several recent achievements in this domain, concentrating on asymptotic normality andfunctional central limit theorems.

In work on the Riemann zeta function, it is of interest to evaluate certain integrals involving the characteristic polynomials of N x N unitary matrices and to derive asymptotic expansions of these integrals as N -> \infty. In this talk, I will obtain exact formulas for several of these integrals, and relate these results to conjectures about the distribution of the zeros of the Riemann zeta function on the critical line. I will also explain how these results are related to multivariate statistical analysis and to the hypergeometric functions of Hermitian matrix argument.

In this talk, we resolve several questions related to a certain heavy traffic scaling regime (Halfin-Whitt) for parallel server queues, a family of stochastic models which arise in the analysis of service systems. In particular, we show that the steady-state queue length scales like $O(\sqrt{n})$, and bound the large deviations behavior of the limiting steady-state queue length. We prove that our bounds are tight for the case of Poisson arrivals. We also derive the first non-trivial bounds for the steady-state probability that an arriving customer has to wait for service under this scaling. Our bounds are of a structural nature, hold for all $n$ and all times $t \geq 0$, and have intuitive closed-form representations as the suprema of certain natural processes. Our upper and lower bounds also exhibit a certain duality relationship, and exemplify a general methodology which may be useful for analyzing a variety of stochastic models. The first part of the talk is joint work with David Gamarnik.