Progress in Steinís Method
(5 Jan Ė 6 Feb 2009)
One of the greatest achievements of
probability has been its success in approximating the
distributions of arbitrarily complicated random processes in
terms of a rather small number of Ďuniversalí processes ó
Brownian motion, the Poisson process, the Ewens sampling
formula, Airy processes, stochastic Loewner evolution and so
on. The standard approach is to consider sequences of
processes, indexed by a parameter n, and to establish that
suitably normalized versions of the processes converge in
distribution to one of the standard processes as n tends to
infinity. However, for practical purposes, it is much more
important to know how accurate such an approximation is for
a particular process with a fixed value of n, and this is a
more difficult question to answer. For instance, central
limit theorems were known already around 1715, and in full
generality by 1900, whereas the corresponding approximation
theorem of Berry and Esseen was only proved in 1941. Steinís
method, introduced in 1970, offers a general means of
solving such problems. By constructing and exploiting a
novel characteristic operator associated with a random
system ó most often, the one used as the approximation ó it
turns out to be possible to make precise assessments of the
approximation error in a wide variety of circumstances.
Steinís original application was in the context of central limit approximation to partial sums of random variables having a stationary dependence structure, a problem involving the normal distribution and the real line. However, his method has a big advantage over most other techniques in that it can in principle be used for approximation in terms of any distribution on any space, including random variables on the real line, processes on a space of sequences, functions or measures, and combinatorial structures on discrete spaces. A further big advantage over its competitors is that strong independence requirements are not needed to make the method work (though they may of course simplify many arguments and the form of the bounds that can be attained). As a result of this considerable freedom, its uses have proliferated, with approximations not only to the normal distribution, but also to the Poisson distribution, to multivariate normal distributions, to diffusions, to Poisson processes, to the Ewens sampling formula, to the Wigner semi-circle law, and more. The proceedings of a workshop in the program ĎSteinís method and applications: a program in honor of Charles Steiní, held at the Institute for Mathematical Sciences of the National University of Singapore in August 2003, illustrate the variety and richness of the field.
In the five years since that program was held, and stimulated to a considerable degree by the impulse that it provided, there have been a number of significant new developments. The first of these concerns large deviations and concentration inequalities. A second new field is the application of Steinís method to problems having an essentially algebraic component. A third area is the combination of Stein's method with Malliavin calculus to prove bounds in normal and gamma approximation of functionals of infinite-dimensional Gaussian fields. A fourth area in which the method is finding increasing application is that of random geometrical graphs. All these new developments have opened up new ways to a wider range of applications of Steinís method.
In view of the breadth and diversity of these and other recent advances, the time is now ripe to hold a further program, with the aim of bringing together the people actively involved in the area, and of cementing and further promoting the development of the field. In addition to the general scientific aim, program is also designed to develop research in Steinís method in Southeast Asia, where there is a growing interest in the method. It also aims, by way of a series of tutorial lectures, to encourage more young mathematicians to undertake research in the field.
(a) Poisson approximation, by Andrew Barbour
(b) Normal approximation, by Larry Goldstein
(c) Malliavin calculus and related topics, by Giovanni Peccati
(d) Applications in algebra, by Jason Fulman
(e) Stein's method in concentration inequalities, spin glasses, random matrices, and strong approximations, by Sourav Chatterjee
Date: 28 Jan 2009
Venue: IMS Seminar Room (House 3)
For attendance at these activities, please complete the online registration form.
The following do not need to register:
Those invited to participate.
Those applying for membership with financial support.
The Institute for Mathematical Sciences invites applications for membership for participation in the above program. Funds to cover travel and living expenses are available to young scientists. Applications should be received at least three (3) months before the commencement of membership. Application form is available in (MSWord|PDF|PS) format for download.