STEIN’S METHOD AND APPLICATIONS:
A PROGRAM IN HONOR OF CHARLES STEIN
(28 July  31 August 2003)
~ Abstracts ~
Normal approximation for sums of
i.i.d. random variables
Charles Stein, Stanford University
Using an identity of Bolthausen [1984] based on an
exchangeable pair, I shall develop some identities related to
the BerryEsseen theorem for sums of independent, identically
distributed random variables.
[1984] An estimate of the remainder in a combinatorial
central limit theorem. Zeitschrift f"ur
Wahrscheinlichkeitstheorie und Verwandte Gebiete, 66, 379386.
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Parameter estimation for highly
structured systems
Adrian Baddeley, University of Western Australia
A highly structured stochastic system, such as a Markov
random field or conditional independence model, can be
represented as the stationary distribution of a Markov chain.
Typically this representation is used as the basis of Markov
Chain Monte Carlo methods for the system. However, it can also
be used for Stein's method. Furthermore, the generator of this
chain is a readymade source of unbiased estimating equations
for the model parameters. Special cases include maximum
likelihood, maximum pseudolikelihood, the TakacsFiksel method
for spatial point processes, and the reduced sample estimator
of lifetime distribution from rightcensored data.
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Convergence rate for CLT of maxima in
cubes
Zhidong Bai, National University of Singapore
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Some applications of Stein's
method
A. D. Barbour, Universitaet Züerich
Stein's method has proved to be an extremely versatile tool
for establishing distributional approximations. In this talk,
we describe three settings of apparently quite different
kinds, in which Stein's method has nonetheless turned out to
play a significant part in the analysis: `small world'
networks, (logarithmic) combinatorial structures and
biological metapopulation models. In each case, Stein's method
emerged by accident, rather than by design, illustrating its
widespread usefulness.
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Mathematical modelling of parasitic
diseases
A. D. Barbour, Universitaet Züerich
Mathematical arguments have been used to help
in fighting disease ever since the work of Daniel Bernoulli in
the early 1700’s. A common theme nowadays is to use
mathematical models to predict the efficacy of competing
public health measures, such as various immunization
strategies, in relation to their probable cost. In this talk,
the speaker illustrates that modelling can be effective even
at a more basic level. A pair of simple differential equations
was used to make a discovery about the natural history of a
widespread parasitic infection.
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Stein's method and Palm distributions
Tim Brown, Australian National University
The probabilistic approach to Stein's method for
distributions on the nonnegative integers (using reversible
Markov birthdeath processes) generalises to point processes.
It extends to what are often called mixed sample processes,
processes in which the number of points on the whole space has
an arbitrary distribution . Conditional on this number the
points are distributed in an independent and identical way.
These processes include the Poisson, binomial and negative
binomial on compact spaces. Such processes are simple examples
of a general class called polynomial birthdeath processes.
The general class can be used to refine the Poisson
approximation to a binomial process using probability
distributions in contrast to expansions using Stein's method
which are typically not approximations by probability
distributions. There are interesting connections for these
processes with the point process theory of Palm distributions,
all this brought into being from Stein's method.
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Stein’s method, concentration
inequalities and normal approximation
Louis Chen, National University of Singapore
Concentration inequalities in normal approximation using
Stein’s method correspond to the smoothing lemma in the
Fourier analytic method. They provide bounds on the
discrepancy between a distribution and its translations. The
discrepancy could be Kolmogorov or total variation distance.
We discuss the role such inequalities play in two kinds of
normal approximation using Stein’s method. The first concerns
uniform and nonuniform bounds on the difference between the
distribution function of a sum of dependent random variables
and the standard normal distribution function. The second,
which uses coupling, is about normal approximation in some
kind of total variation for sums of independent integervalued
random variables.
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Applications of Stein’s method in
appearances of attributes
Ourania Chryssaphinou, University of Athens
Generalizing the concept of attribute as it has been
analyzed by W. Feller we study the number of renewal
appearances of general attributes. Using Stein’s method we
obtain limit results of Poisson type under quite general
conditions. We present certain applications concerned known
models, their generalization and new ones.
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Random variables defined on
random binary search trees
Luc Devroye, McGill University
Several random variables defined on random binary search
trees have normal limit laws that can be obtained by Stein's
method. We investigate in particular sums of functions of
subtrees, and obtain universally applicable laws. Special
cases include the number of leaves, the number of subtrees
with a given pattern, the product of the sizes of all subtrees,
and the sum of the heights of all subtrees.
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Historical roots of Stein's
method
Persi Diaconis, Stanford University
Stein's Method developed over many years. I will examine
forerunners such as Lindeberg's approach, Stein's early efforts
using characterisitic functions for the combinatorial limit
theorem and end with a review of original paper.
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The Search for randomness
(Public lecture)
Persi Diaconis, Stanford University
The speaker will discuss some of our most primitive examples
of random phenomena: tossing a coin, rolling dice and shuffling
cards. While common practice can produce randomness, usually a
close look shows that it just isn't so.
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Stein's method for Gibbs
measures
Peter Eichelsbacher, Fakultaet fuer Mathematik, RuhrUniversitaet
Bochum, Germany
Stein's method provides a way of finding approximations to
a distribution of a random variable and gives at the same time
estimates of the approximation error involved. We provide
Stein's method for the class of Gibbs measures with a
density e^{V} where V is the energy
function. Combining the probabilistic view of Stein's method
introduced by Barbour and the observation from Preston that
Gibbs states are invariant measures of certain timereversible
spatial birth death processes we introduce the Stein equation
and give Stein bounds. Using size bias couplings, we treat an
example of Gibbs convergence for strongly correlated random
variables due to Chayes and Klein.
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Stein's method, Poisson and
compound Poisson approximations (Tutorial lectures)
Torkel Erhardsson, Royal Institute of Technology, Stockholm
We give an overview of the use of Stein's method for
Poisson approximation (lectures 12) and compound Poisson
approximation (lectures 34). We shall focus on basic results:
the Stein equations and properties of their solutions (in
particular the socalled magic factors), and how these are
used to construct explicit error bounds for Poisson or
compound Poisson approximations of sums of nonnegative integer
valued random variables (the local and coupling approaches,
monotone couplings, etc.) We give proofs for the most
important results, as well as examples of applications. We
shall also discuss some generalizations of the basic results,
including PoissonCharlier expansions, signed compound Poisson
measure approximation, and compound Poisson approximation as a
byproduct of Poisson process and compound Poisson process
approximation.
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Stein's method, Markov renewal
point processes and strong memoryless times
Torkel Erhardsson, Royal Institute of Technology, Stockholm
Let W be the number of points in (0,t] of a
stationary finitestate Markov renewal point process with
marks falling into a rare subset B of the state space
S. We want to find a good approximating compound
Poisson distribution for L(W) and an explicit bound for
the approximation error. We present a partial solution to the
problem, obtained by expressing W as an integral with
respect to an embedded renewalreward process, and using
Stein's method, Palm theory for point processes, and
couplings. For the error bound to be small, S should
contain at least one frequently visited state. We show how to
embed the Markov renewal point process into another such
process which has a frequently visited state, using auxiliary
random variables called strong memoryless times. As an
illustrative example, we consider the number of points in (0,t]
of a stationary renewal process.
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Normal approximation for hierarchical
sequences
Larry Goldstein, University of Southern California
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Asymptotic spectral distributions
via Stein equations
Friedrich Götze, Bielefeld University
We show the asymptotic Wigner distribution of eigenvalues
of random matrices distributed according to a martingale type
stochastic model for the coefficients, which includes the
WignerEnsemble. We use a Stein equation with singularities at
the boundary of the compact support to prove the result and
investigage some applications. Further possible applications
of Stein's method will be discussed.
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Concise rates of convergence in the
Studentised central limit theorem
Peter Hall, Australian National University
The central limit theorem for the Studentised mean has been
of interest since Gosset first considered randomly normalised
statistics almost 100 years ago. Early work, for example based
on Edgeworth expansions, was conducted under the assumption of
sufficiently many finite moments. Much more recently it has
been been shown that moment conditions do not directly affect
the validity of the central limit theorem for Student's t
statistic. Indeed, the Studentised mean has an asymptotic
standard normal distribution if and only if the sampling
distribution is in the domain of attraction of the normal law.
Assuming only this minimal condition we discuss leadingterm
expansions of the distribution of the Studentised mean. Under
additional conditions these expansions approximate those of
Edgeworth type. They give concise rates of convergence in the
central limit theorem.
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Comparing stationary distributions
of Markov Chains through Stein's method
Susan Holmes, Stanford University
I will show how Stein's method can be used to compare
distances between distributions. Starting with tunable birth
and death chains, I will show how we can choose the
exchangeable pair and then compare the Stein operators, thus
providing useful bounds on distances between distributions for
which we don't necessarily know the Stein equation to start
with.
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Characterization of Brownian motion
on manifolds – an infinitedimensional extension of Stein’s
idea
Elton P. Hsu, Northwestern University
We explain how Stein’s characterization of the standard
normal random variable is related to an integration by parts
formula in the path space over a Riemannian manifold.
This relation motivates the following result: Brownian
motion on a Riemannian manifold is uniquely characterized by
the integration by parts property for the gradient operator.
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The central limit theorem for the
independence number for minimal spanning trees on random
points in the unit square
Sungchul Lee, Yonsei University
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Fixeddomain asymptotic for a
subclass of Matérntype Gaussian random fields
WeiLiem Loh, National University of Singapore
Michael Stein (1989) proposed the Matérntype Gaussian
random fields as a very flexible class of models for computer
experiments. This talk considers a subclass of these models
that are exactly once mean square differentiable. In
particular, the likelihood function is determined in
closedform and under mild conditions, the sieve maximum
likelihood estimators for the parameters of the covariance
function are shown to be weakly consistent with respect to
fixed domain asymptotics.
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Random geometric graphs and
random directed graphs
Mathew Penrose, University of Durham
Random geometric graphs are a natural alternative to
classical random graph models, in which n vertices are
independently randomly scattered in ddimensional space
(typically, uniformly over the unit square with d=2), with an
edge included between nearby points. We consider topics such
as giant component, clique number, and number of appearences
of a given graph as a subgraph. These graphs are the subject
of a recent monograph by the speaker.
If time permits, we shall also discuss recent work with
Andrew Wade on directed nearestneighbour graphs. In both
cases, Stein's method is a useful tool.
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Stein's method for chisquare
distributions, law of large numbers, and discrete
distributions from a Gibbs view point (Tutorial lectures)
Gesine Reinert, University of Oxford
In these tutorial lectures it shall be illustrated how
Stein's method can be applied for distributional
approximations in more generality. The first lecture will
treat chisquare approximations, giving an application of the
socalled generator approach to Stein's method. Afterwards
this approach will be applied to the law of large numbers, and
to convergence of empirical measures. Both the generator
approach and a coupling characterization are put into use in
the next part, considering discrete distributions in general.
Examples, including an epidemic model, will be presented in
the last lecture.
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Application of Stein's method to
word matches in DNA sequences
Gesine Reinert, University of Oxford
Word count statistics are essential in many procedures for
assessing statistical significance in DNA sequence comparison.
Statistical large sample approximations often can only be
used in connection with bounds on the quality of the
approximation. This approach will be applied to the analysis
of the socalled D_{2} test statistic, counting
the number of kword matches between two random
sequences. This statistic is used in BLAST.
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Stein's method and Edgeworth expansions for
sums of dependent variables
Vladimir Rotar, San Diego State
University and the Central Economics and Mathematics Institute
of Russian Academy of Sciences
View/download PDF
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Stein's method and normal
approximation (Tutorial lectures)
QiMan Shao, National University of Singapore and
University of Oregon
We shall give an overview of the use of Stein’s method for
normal approximation. We start with basic results on the Stein
equations and their solutions and then prove several classical
limit theorems to illustrate the beauty of Stein’s method. The
focus will be on the ideas behind different approaches such as
the concentration inequality approach, induction approach and
exchangeable pair approach. We shall also discuss the uniform
and nonuniform BerryEsseen bounds for independent random
variables as well as for locally dependent variables.
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The BerryEsseen bound for the
Student tstatistic via Stein's method
QiMan Shao, National University of Singapore and
University of Oregon
We show how the Stein method can be used to prove the
BerryEsseen bound for the Student tstatistic and for
studentized statistics in general. The approach is based on
some randomized concentration inequalities.
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Aspects of the probability theory
of the MST and TSP
J. Michael Steele, University of Pennsylvania
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Operator inequalities and their
applications
Sergey Utev, University of Nottingham
 How to find a good approximation on the stoploss
premiums of the first and second orders in the individual
risk model with independent claims occurrences?
 How to measure the impact of dependence between claims
occurrences?
 How to derive extremal properties of Rademacher
functions or find the best constants in the Rosenthal
inequality?
The SteinChen method, stochastic orderings and operator
inequalities are employed to answer these and similar
questions.
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Stein's method and Poisson process
approximation (Tutorial lectures)
Aihua Xia, National University of Singapore and University
of Melbourne
We start with an introduction lecture on Poisson processes
on the real line and then Poisson point processes on a locally
compact complete separable metric space. The focus will be on
important properties of Poisson point processes and how to
characterize a Poisson point process. The second part will
review the basics of Markov immigrationdeath processes,
followed by the definition of Markov immigrationdeath point
processes. We then explain how a Markov immigrationdeath
point process evolves, establish its generator and find its
stationary distribution. The final part concentrates on how to
use the knowledge of Markov immigrationdeath point processes
to construct Stein's method for Poisson process approximation
with various applications.
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Stein's method for compound Poisson
approximation via immigrationdeath processes
Aihua Xia, National University of Singapore and University
of Melbourne
View/download PDF
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Central limit theorems in
geometric probability
Joseph E. Yukich, Lehigh University
We establish general central limit theorems for measures
induced by binomial and Poisson point processes in
ddimensional space. The limiting Gaussian field has a
covariance structure which depends on the density of the point
process. The general results are applied to deduce weak
convergence of measures induced by random graphs (minimal
spanning tree, nearest neighbor, Voronoi, and sphere of
influence graph), random sequential packing models (ballistic
deposition and spatial birth growth models), the process of
maximal points of a random sample, and as well as to measures
induced by spacing statistics. The talk is based on joint work
with Y. Baryshnikov and M. Penrose.
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A central limit theorem and large
deviation principle for decomposable random variables
Martin Raič, University of Ljubljana Slovenia
We present a central limit theorem with nonuniform bounds
for sums of dependent random variables. The decay is
sufficiently fast to allow us to derive large deviation
principles. Our result is based on the concept of decomposable
random variables, introduced by Barbour, Karoński and Ruciński.
This approach to dependence has turned out to be applicable in
various contexts, such as local dependence, random graphs and
random permutations.
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Dependent superpositions of
point processes
Dominic Schuhmacher, University of Zürich
We compare the distribution of a (possibly dependent)
superposition of point processes to a Poisson process
distribution. Using the method, we obtain upper bounds on the
d_2 distance (a Wasserstein distance) between these
distributions.
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