# Probability Seminar

## Topics

September 25, 1997
Ted Hill
Georgia Tech

#### Five corollaries in need of a theorem

Four different results remarkably similar to Levy's classical continuity theorem will be described, including for each a basic definition, representation theorem, continuity theorem, and examples. What is missing is the general analytical result from which these all must follow. Audience insights are what I am hoping for!

October 2, 1997
Jim Dai
Georgia Tech

#### Queueing networks in heavy traffic: current status

Queueing networks have been used to model complex manufacturing and telecommunication systems. Most of these networks are beyond exact mathematical analysis. Such a network is said to be in heavy traffic if the total loads imposed on one or more of its processing resources (servers) are approximately equal to the capacities of those resources. The heavy traffic regime is of interest not only because many real systems are characterized by approximate balance between resource capacities and workload inputs, but also because in this regime the qualitative nature of system behavior and the structural features of good control policies show up in sharpest relief.

A fundamental question is: When can one approximate the normalized vector process of inventory levels for various customer classes by a multi-dimensional reflected Brownian motion under heavy traffic conditions? This talk will survey the latest developments in the area, including recent work of Bramson and Williams. No prior knowledge of queueing networks is needed.

October 9, 1997
Rafal Latala
Georgia Tech and Warsaw University

#### Estimation of moments of sums of independent random variables

For the sum $S=\sum X_{i}$ of a sequence $(X_{i})$ of independent mean zero (or nonnegative) random variables, we give lower and upper estimates of moments of $S$. Estimates are exact, up to some universal constants and extend the previous results for particular types of variables $X_{i}$.

October 16, 1997
Miguel Arcones
University of Texas, Austin

#### Limit theorems under long range dependence conditions

In this talk several limit theorems over a sequence of stationary Gaussian sequence of random vectors will be considered. Let $\{X_{j}\}_{j=1}^{\infty}$ be a stationary Gaussian sequence of random vectors with mean zero. We present some new results on the convergence in distribution of $$a_{n}^{-1}\sum_{j=1}^{n}(G(X_{j})-E[G(X_{j})]),\leqno{(1)}$$ where $G$ is a real function in $R^{d}$ with finite second moment and $\{a_{n}\}$ is a sequence of real numbers converging to infinity. We give necessary and sufficient conditions for the weak convergence of (1) for all functions $G$ with finite second moment. These conditions allow us to obtain distributional limit theorems for general sequences of covariances. These covariances do not have to decay as a regularly varying sequence nor be eventually nonnegative (as in Taqqu, 1975, 1977; and Dobrushin and Major, 1979). The convergence in distribution of (1) is determined by the first two terms in the Fourier expansion of $G(x)$. The explicit form of characteristic function of the limit of (1) is given.

We also give some sufficient conditions for the compact law of the iterated logarithm for $$(n2 \log \log n)^{-1/2}\sum_{j=1}^{n}(G(X_{j})-E[G(X_{j})]).\leqno{(2)}$$ Our result builds on Ho (1995), who proved an upper--half of the law of iterated logarithm for (2).

October 23, 1997
Michael Maxwell
Georgia Tech

#### A central limit theorem for Markov chains

A central limit theorem and invariance principle is obtained for additive functionals of a stationary ergodic Markov chain, say S_n=g(X_1)+...+g(X_n), where Eg(X_1)=0 and Eg(X_1)^2 is finite. The conditions imposed restrict only the growth of the conditional means E(S_n|X_1). No other restrictions on the dependence structure of the chain are required.

October 30, 1997
Don McNickle
University of Canterbury

#### Two applications of Queueing Theory

Queueing Theory is often derided as highly theoretical and impractical. Yet every so often it works surprisingly well. These are two such cases:

The Kinleith Weighbridge
Every year some 2.5 million of wood are transported to the Carter Holt Harvey complex at Kinleith. More than 400 truck-loads per day pass over a single lane weighbridge. Waiting times had been observed to sometimes reach 20 minutes per truck. A second weighbridge in parallel was under consideration. The queueing model had to deal with non-stationary behaviour and a move-up effect as the trucks moved on to the bridge.

Revising G1/AS1
Works Consultancy Services was contracted by the Buildings Industry Authority to undertake a study of the number of sanitary facilities to be provided in buildings, leading to revision of the G1/AS1 tables in the New Zealand Building Code. A very extensive data-gathering exercise to predict demand and occupancy times for various kinds of buildings was carried out. Simple queueing models proved to be adequate for predicting the waiting time distributions that the new standards would produce. The potential savings of the new standards were surprising.

November 6, 1997
Georgia Tech

#### A survey of results on the hard-core lattice gas model

The hard-core lattice gas model was introduced as a simple model for a gas consisting of particles with nonnegligible radii, as a consequence of which any valid configuration forbids two neighboring sites (on, say, a square lattice) from being occupied at the same time.

In graph-theoretic terms, such valid configurations correspond to independent sets (a.k.a. stable sets). Given a finite (or countably infinite) graph, and an arbitray real $\lambda$, the model associates a weight $\lambda^{|I|}$ to each independent set $I$ of the graph. This model is of interest and significance in statistical physics, theory of computing, and communication networks. The speaker intends to review some of the recent results and open problems concerning the hard-core model.

In particular, the notions of Dobrushin-Shlosman (spatial) mixing conditions and rapid mixing of certain reversible dynamics and other mixing conditions which guarantee the uniqueness of a hard-core infinite-volume (Gibbs) distribution will be explained.

November 13, 1997
Bob Foley
Georgia Tech

#### Join the shortest queue: stability and exact asymptotics

Joining the shortest queue is a simple discipline to describe, but difficult to analyze. In this paper, we determine the stability conditions and exact asymptotics for the total number of customers in the system reaching some large level. We determine the stability conditions for a system consisting of m queues each served by a single exponential server. Each subset of the m queues has its own Poisson arrival process which chooses the shortest queue among that subset of queues. We will determine exact asymptotics for the expected time from an empty system until the total number of customers reaches a large level. We will also be able to determine exact asymptotics for the stationary distribution \pi.

November 20, 1997
Ron Fox
Physics, Georgia Tech

#### Rectified Brownian Movement in Molecular and Cell Biology

A unified model is presented for rectified Brownian movement as the mechanism for a variety of putatively chemo-mechanical energy conversions in molecular and cell biology. The model is established by a detailed analysis of ubiquinone transport in electron transport chains and of allosteric conformation changes in proteins. It is applied to P-type ATPase ion tranporters and to a variety of rotary arm enzyme complexes. It provides a basis for the dynamics of actin-myosin cross-bridges in muscle fibers. In this model, metabolic free energy does no work directly but instead biases boundary conditions for thermal diffusion. All work is done by thermal energy which is harnessed at the expense of metabolic free energy through the establishment of the asymmetric boundary conditions.

December 4, 1997
Christian Houdre
Math, Georgia Tech

#### Some Jackknife--Based Deviation Inequalities

January 15, 1998
Bartek Blaszczyszyn
University of Wroclaw

#### Factorial Moment Expansion for Functionals of Point Processes With Application to Approximations of Stochastic Models

For a given functional of a simple marked point process we find an expression that is analogous to a finite expansion of the probability generating functional with respect to factorial moment measures. The terms of the expansion are integrals of some real functions with respect to factorial moment measures of the point process. The remainder term is an integral of some functional with respect to a higher order Campbell measure. The expansion is also related to Palm--Khinchin formulas.

We use this result as a basis for a new approach to light traffic approximations of stochastic systems, which can be viewed as being driven by a simple marked point process.

In a special case of thinning a point process, we obtain Taylor approximations. Particular results for Petri nets, ruin functions of risk models, workload vectors of many-server queues in a Markov modulated environment and the clump size i the Boolean model are presented.

The talk is based on the Ph.D. thesis of the author and some complementary studies.

January 22, 1998
David McDonald
University of Ottawa

#### Do birds of a feather flock together (Testing for a nonhomogeneous Poisson process)

Redwinged blackbirds nest year after year in the Mer bleue conservation area near Ottawa. Since the topography of the swamp doesn't change much from year to year we can observe a sequence of independent realizations of the spatial point processes of nest locations. If a bird chooses the location of its nest according to topological criteria (best known to birds but which certainly avoid the middle of ponds) independently of the location of the nests of other redwings then one might describe the location of nests in a given year as a realization of a nonhomogeneous Poisson process.

We propose a test for Poissoness which involves a characterization of Poisson processes and Cramer-Von Mises statistics. We also discuss extensions of these ideas to nonparametrique quality control and nonparametrique regression.

January 29, 1998
John Elton
Georgia Tech

#### Convex Domination of Probability Measures

An elementary geometric proof is given for the basic representation for convex domination of measures in the case of purely atomic measures with a finite number of atoms. Then a simple argument is given to show how this implies the general case, including both probability measures and finite Borel measures, on infinite dimensional spaces. The infinite-dimensional case follows quickly from the finite-dimensional case by using Grothendieck's approximation property.

February 5, 1998
Y. L. Tong
Georgia Tech

#### The Role of the Covariance Matrix in the Least-Squares Estimation for a Common Mean

For $n>1$ let $\bX=(X_1,\dots , X_n)'$ have a mean vector $\theta\bone$ and covariance matrix $\sigma^2\bsigma$, where $\bone = (1,\dots, 1)'$, $\bsigma$ is a known positive definite matrix, and $\sigma^2>0$ is either known or unknown. This model has been found useful when the observations $X_1,\dots, X_n$ from a population with mean $\theta$ are not independent. In this talk we show how the variance of $\hat\theta$, the least-squares estimator of $\theta$, depends on the covariance structure of $\bsigma$. Specifically, we give expressions for Var$(\hat\theta)$, obtain its lower and upper bounds (which involve only the eigenvalues of $\bsigma$), and show how the dependence of $X_1,\dots, X_n$ plays a role on Var$(\hat\theta)$. Examples of applications are given for $M$-matrices, for exchangeable random variables, for a class of covariance matrices with a blockwise correlation structure, and for the Danish twin data.

February 12, 1998
Carlangelo Liverani
University of Rome, Italy

#### Central Limit Theorem for Dynamical Systems

The aim of the talk is to present a simple approach to obtaining the Central Limit Theorem (CLT) for hyperbolic dynamical systems. The method is inspired to techniques developed for the study of the CLT in reversible Markov processes. The results are particularly suitable for application to concrete examples.

February 19, 1998
Philippe Barbe
Math, Georgia Tech and CNRS, France

#### A Sharp Petrov Type Large Deviation Formula

Let $X, X_i,i\geq 1$ be a sequence of independent and identically distributed random vectors in $\RR^d$. Consider the partial sum $S_n:=X_1+\ldots +X_n$. Under some regularity conditions on the distribution of $X$, we obtain an asymptotic formula for $P(S_n\in nA)$ where $A$ in an ARBITRARY Borel set.

The talk is based on a joint work with Michel Broniatowski (Universit\'e de Reims).

February 26, 1998
Tito Manlio Homem de Mello
ISyE, Georgia Tech

#### Estimation of Derivatives of Nonsmooth Performance Measures in Regenerative Systems

We investigate the problem of estimating derivatives of expected steady-state performance measures in parametric systems. For the class of regenerative Markovian systems we show that, under some assumptions, the process of directional derivatives at a fixed point regenerates together with the original process. An example illustrates that the imposed conditions must be more strict than in the differentiable case. Besides yielding an estimation procedure for directional derivatives and subgradients of equilibrium quantities, the result allows us to derive necessary and sufficient conditions for nondifferentiability of the expected steady-state function. As a by-product of the analysis we provide a limit theorem for more general compact-convex multivalued processes.

March 5, 1998
Sergey Bobkov
Math, Georgia Tech

#### Log-Sobolev Inequality for Poisson Measure and its Application to Large Deviations

Log-Sobolev inequalities are known to be a powerful tool in the study of Gaussian tails of Lipschitz functionals on product spaces. We will show what kind of log-Sobolev inequality holds for Poisson measure and how it can be used to recover Poisson tails for Lipschitz functionals. The talk is based on a recent paper with M. Ledoux.

March 12, 1998
Soren Asmussen
Lund University

#### Large Deviations Type Results in the Presence of Heavy Tails

We study stochastic processes with negative drift and positive jumps which are subexponential (for example, Pareto, lognormal or Weibull with decreasing failure rate). Intuitively, large deviations for such a process occurs because of one big jump.

For a random walk or Levy process, we identify the large deviations path leading to exceedance of a large level. In particular, it is shown that the time of exceedance, when properly normalized, has a limit which is either Pareto or exponential.

For a reflected random walk, we find the tail asymptotics of the maximum during the cycle. This determines the extreme value behaviour of the process as a whole. The results yield as a corollary similar results for storage processes with state dependent release and asymptotics for their stationary distribution. By sample path duality, this extends to approximations for finite or infinite horizon ruin probabilities in insurance risk models with compounding assets or a more general state--dependent premium rule.

March 19, 1998
Sabine Schlegel
University of Ulm

#### Ruin Probabilities in Perturbed Risk Models

We consider the asymptotical behaviour of the ruin function in perturbed and unperturbed non-standard risk models when the initial risk reserve tends to infinity. We give a characterization of this behavior in terms of the unperturbed ruin function and the perturbation law provided that at least one of the two is subexponential. Through a number of examples for the claim arrival process as well as the perturbation process we show that our result is a generalization of previous work on this subject.

April 2, 1998
Daryl J. Dayley
School of Math Sciences, Australian National University

#### Long Range Dependence of Point Processes

The long range dependence (LRD) of a point process can be defined in terms of intervals between points (Long Range interval Dependence, i.e. LRiD) or of the counting function N(.) (Long Range count Dependence, i.e. LRcD). Both definitions are given and illustrated: for example, a renewal process can never be LRiD but can be LRcD.

The transformation of one point process into another is examined. Any LRcD property is shown to be invariant under the independent translation of points. Such a transformation is equivalent to feeding a point process through an infinite server queue. Other queueing transformations are examined and shown to be capable of preserving LRcD properties, and of introducing them.

(Includes joint work with Rein Vesilo of Macquarie University, Sydney.)

April 9, 1998
Dick Serfozo
ISyE, Georgia Tech

#### Departure Processes in Long Tandem and Treelike Networks are Asymptotically Poisson

Consider a tandem network of n nodes in which service times at each node are independent exponentially distributed with a common rate. Let D_n denote the point process of departure times from node n, where D_n(I) is the number of departures in the time interval I. It was recently proved by Prabhakar and Mountfort that D_n converges in distribution to a Poisson process as n tends to infinity. This talk gives a simpler proof of this result and shows how it extends to multi-server queues and treelike networks. The proof is based on a coupling argument that uses a special representation of such processes and ideas from thinnings of point processes.

April 16, 1998
Carl Spruill
Math, Georgia Tech

#### Asymptotically Optimal Designs for Testing in Logistic Regression

Optimal designs for testing hypotheses about a single parameter in a multiparameter exponential family are presented and applied in the setting of logistic regression. The designs are optimal in the sense that they maximize the Pitman efficiency in the class of asymptotically locally most powerful unbiased tests. The familiar Hoel-Levine designs are optimal for logistic regression under the assumption that the explanatory variables have no effect on the probability of a favorable response.

April 23, 1998
Otis Jennings
ISyE, Georgia Tech

#### Stability of ell-Limited Polling Policy

We consider multiclass queueing networks with setups. It is known that many setup policies, including the Setup Avoidance Policy, are inefficient in the sense that the bottleneck machines are underutilized no matter how high the work-in-process is. We study a family of ell-limited polling setup policies. Using a fluid limit model approach, we show that these policies are efficient in stochastic multiclass queueing networks, no matter how long the setup times are. Our result generalizes the one by Kumar and Seidman (1990) for deterministic networks.

April 30, 1998
Lisa Bloomer
Math, Georgia Tech

#### Constructing Natural Random Probability Measures

This will be an overview of several different methods of constructing random probability measures in a natural way. These methods include the Dirichlet process, choosing moment sequences, choosing barycenter arrays, and the Dubins and Freedman method of choosing distribution functions. Known results and open problems will be presented.

Last updated: April 27, 1998 by J. Hasenbein ( johnny@isye.gatech.edu)