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
Presad Tetali
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)