Probability Seminar

Topics



January 9, 1996
D. Nualart
University of Barcelona

Regularity of probability laws via Malliavin Calculus

As an application of the Malliavin calculus we will present some criteria for the existence and smoothness of the density of a Wiener functional. We will show that the support of the law of a weakly differentiable random variable is connected, and we will discuss the positivity of its density in the interior of the support. The particular example of diffusion processes will be analyzed.


January 18, 1996
Jim Dai
Math/ISyE
Georgia Institute of Technology

Performance and Scheduling of Processing Networks

On Saturday, February 24, Professor P. R. Kumar from University of Illinois will give two talks on "Scheduling Manufacturing Systems: Performance Analysis and Design". His talks are part of a workshop organized by the Center for Applied Probability at Georgia Tech. To help general audience better prepared for his talks, I will give an introductory talk on multiclass queueing networks. A number of topics, including model definition, stability, performance bounds and scheduling will be discussed.


January 25, 1996
Dan Mauldin
Department of Mathematics
University of North Texas

Random Distributions and Homeomorphisms

Several schemes for producing random probability measures or distributions supported on the unit interval are presented. These distrubtions can also be thought of as autohomeomorphisms of the unit interval. We will present some basic properties of these random objects and illustrate how some properties were guessed via computer simulations..


February 2, 1996
David McDonald
Department of Mathematics
University of Ottawa

Rare Events for a Markov Chain

The performance of a telecommunication system is often measured by the frequency or probability of rare events such as a buffer overflow. Such a system is modeled by a Markov chain with a large state space and stationary distribution pi. A rare event corresponds to an excursion to a forbidden set F with small stationary probability pi(F). The distribution of the duration of such excursions is asymptotically exponential and during each excursion there is a small clump of visits to the forbidden set. The Poisson clumping heuristic due to Aldous connects the meanclump size and the steady state probability of the forbidden set with the mean excursion length.

Here we consider one node in a queueing network whose transitions may be represented by an Markov additive process which recurs to a reflection point at zero. The steady state pi is not known so the clumping heuristic doesn't work. Nevertheless it is possible to perform an h-transformation or {twist} which turns the additive process around much like the one dimensional associated random walk described in Feller Volume II. This twist automatically yields the rough asymptotics of the mean excursion length. The exact asymptotics are given by probabilistic expression involving the twisted Markov additive process. The steady state of the network conditioned on the specified node being in F also follows. The twisted process is transient to F so a computer simulation of the rare event is very quick. The hitting distribution of the twisted process on F immediately yields that of the original Markov chain.


February 8, 1996
M. Borodovsky
Department of Mathematics
Georgia Tech

February 14, 1996
Gideon Weiss
Haifa A process of zigzag lines with Poisson breakpoints


February 22, 1996
Michael Monticino
Department of Mathematics
University of North Texas

Search, Information, and Gambling

Search theory is one of the oldest areas of operations research. Broadly defined, search theory is a mathematical framework for locating an object of interest. Search theory techniques have to been used to locate such high interest objects such as an H-bomb lost off the coast of Spain, wreckage from the space shuttle Challenger, and North Korean invasion tunnels under the Korean DMZ. This talk will provide an introduction to mathematical search theory and make connections to other areas of decision analysis, including information theory and Bayesian sequential allocation problems.


March 1, 1996
Richard Durrett
Department of Mathematics
Cornell University
From the voter model to species area curves


March 8, 1996
Jianqing Fan
Department of Statistics
University of North Carolina

Exploiting hidden structure in high dimensional data analyses

Nonparametric regression techniques is a data-analytic approach to statistical regression problem. However, for processing high-dimensional data, due to the intrinsic difficulty of the curse of dimensionality, some flexible modeling is needed in order to obtain sensible data analyses. Additive regression models have turned out to be a useful statistical tool in the analysis of high dimensional data sets. In this talk, a direct estimation scheme will be introduced. The explicit definition of this estimator makes possible a fast computation and allows an asymptotic distribution theory. We demonstrate that with an appropriate choice of the weight function, the additive components can be efficiently estimated---each additive component can be estimated as well as if the rest of components were known. Application of local linear fits reduces the design related bias. Estimation of parametric components as well as nonparametric components in the additive partially linear models will also be addressed.

The talk will be based on a recent work with Haerdle and Mammen.


April 4, 1996
Jin Feng
Department of Statistics
University of Wisconsin-Madison

Nonlinear Martingale Problems for Large Deviations of Markov Processes

Weak convergence and large deviation convergence are two most important type of convergences in probability. While weak convergence is linear, large deviation is fully nonlinear.

The usual linear martingale problems (a probabilistic approach for studying contraction linear semigroups) provide a powerful tool for characterizing Markov processes, especially in addressing convergence issues: convergence of generators implies the weak convergence of processes.

In this research, parallel results for large deviations will be established. A class of nonlinear martingale problems will be introduced (which corresponds to a class of nonlinear semigroups named after Nisio). The principal result says if the nonlinear generators converge in a sense, then large deviation principle will follow.

It is hoped that this approach will provide a useful tool for answering probabilistic questions using available nonlinear analytical results, as well as providing tools for studying the large deviation princinple for general processes - measure valued, for example.


May 9, 1996
Sigrun Andradottir
Industrial and Systems Engineering
Georgia Tech

Accelerated Regeneration for Markov Chain Simulations

We describe a generalization of the classical regenerative method of simulation output analysis. Instead of blocking a generatedsample path on returns to a fixed return state, a more general scheme to randomly decompose the path is used. In some cases, this decomposition scheme results in regeneration times that are a supersequence of the classical regeneration times. This "accelerated" regeneration is advantageous in several simulation contexts. It is shown that when this decomposition scheme accelerates regeneration relative to the classical regenerative method, it also yields a smaller asymptotic variance of the regenerative variance estimator than the classical method. Several other contexts in which increased regeneration frequency is beneficial are also discussed.
This is joint work with James M. Calvin and Peter W. Glynn.


May 14, 1996
IC 215, 11 am
Sandy Stidham
Industrial and Systems Engineering
University of North Carolina-Chapel Hill

Airline Yield Management with Overbooking, Cancellations, and No-Show

Airlines face the decision problem of determining how best to allocate their inventory of seats among different fare classes to maximize the total revenue when the future demand is uncertain. This is commonly referred to as seat inventory control or yield management. The problem is further complicated by the possibility of cancellations, no-shows, and the practice of overbooking. Previous research has analyzed the problem without these complications and with other restrictive assumptions. We show how to formulate and solve the seat-inventory-control problem with overbooking, cancellations, and no-shows as a problem of optimal control of admissions to a queueing system, exploiting the substantial body of research that exists on this topic. The optimal policy is characterized by state and time-dependent booking limits for each class. We point out by means of numerical examples that some of the commonly accepted notions about airline yield management may not always be true.

1) We show exact conditions under which it will be optimal to accept a lower-fare class and simultaneously reject a higher-fare class. (A possible scenario which can cause this phenomenon is if the full-fare class is fully refundable and a lower-fareclass is non-refundable.) In this case it does not make sense to have nested booking limits based on fares alone.
2) The booking limits may not be monotonically increasing in the time remaining until the flight.
3) With the possibility of cancellations, an optimal policy may depend on the total capacity as well as the capacity remaining. In fact, for a given capacity remaining the booking limits are monotone (non-decreasing) in the total capacity.

Finally, by pointing out the connection between yield management and queueing control we hope to stimulate further research that exploits this connection to solve additional problems in yield management.


Unless otherwise noted, the seminar meets Thursdays at 3 PM in 269 Skiles. For further information, contact Ted Hill (hill@math.gatech.edu, (404)853-9111)

Last updated: May 13, 1996 by J. Shear ( joannas@isye.gatech.edu)