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STEIN’S METHOD AND APPLICATIONS:
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Probability theory in the first half of the twentieth century was substantially dominated by the formulation and proof of the classical limit theorems --- laws of large numbers, central limit theorem, law of the iterated logarithm --- for sums of independent random variables. The central limit theorem in particular has found regular application in statistics, and forms the basis of the distribution theory of many test statistics. However, the classical approach to the CLT relied heavily on Fourier methods, which are not naturally suited to providing estimates of the accuracy of limits such as the CLT as approximations in pre-limiting circumstances, and it was only in 1940 that Berry and Esseen, by means of the smoothing inequality, first obtained the correct rate of approximation in the form of an explicit, universal bound. Curiously enough, the comparable theorem for the conceptually simpler Poisson law of small numbers was not proved until 26 years later, by LeCam.
These theorems all concerned sums of independent random variables. However, dependence is the rule rather than the exception in applications, and had been increasingly studied since 1950. Without independence, Fourier methods are much more difficult to apply, and bounds for the accuracy of approximations become correspondingly more difficult to find; even for such frequently occurring settings as sums of stationary, mixing random variables or the combinatorial CLT, establishing good rates seemed to be intractable.
It was into this situation that Charles Stein introduced his startling technique for normal approximation in 1972. Now known simply as Stein's method, the technique relies on an indirect approach, involving a differential operator and a cleverly chosen exchangeable pair of random variables, which are combined in almost magical fashion to deliver explicit estimates of approximation error, with or without independence. This latter feature, in particular, has led to the wide range of application of the method.
Stein originally applied his method to give bounds for the accuracy of the CLT for sums of stationary, mixing random variables, but the scope of his discovery has since expanded rapidly. Poisson approximation was studied in 1975; an error bound in the combinatorial CLT was obtained in 1984; the method was extended to the approximation of the distributions of whole random processes in 1988; its importance in the theoretical underpinning of molecular sequence comparison algorithms was recognized in 1989; rates of convergence in the multivariate CLT were derived in 1991; good general bounds in the multivariate CLT, when dependence is expressed in terms of neighborhoods of possibly very general structure, were given in 1996; and Stein's idea of arguing by way of a concentration inequality was developed in 2001 to a point where it can be put to very effective use.
Despite the progress made over the last 30 years, the reasons for the effectiveness of Stein's method still remain something of a mystery. There are still many open problems, even at a rather basic level. Controlling the behavior of the solutions of the Stein equation, fundamental to the success of the method, is at present a difficult task, if the probabilistic approach cannot be used. The field of multivariate discrete distributions is almost untouched. There is a relative of the concentration inequality approach, involving the comparison of a distribution with its translations, which promises much, but is at present in its early stages. Point process approximation, other than in the Poisson context, is largely unexplored: the list goes on.
Due to its broad range of application, Stein's method has become particularly important, not only in the future development of probability theory, but also in a wide range of other fields, some theoretical, some extremely practical. These include spatial statistics, computer science, the theory of random graphs, computational molecular biology, interacting particle systems, the bootstrap, the mathematical theory of epidemics, algebraic analogues of probabilistic number theory, insurance and financial mathematics, population ecology and the combinatorics of logarithmic structures. Many, in their turn, because of their particular structure, have led to the development of variants of Stein's original approach, with their own theoretical importance, one such being the coupling method
This program aims to re-focus interest on understanding the essence of the method and on the various open problems associated with it. It also seeks to foster collaboration in the many fields of application now being studied. Interactions between those involved with different aspects of the method will be an important ingredient in its further development.
There will be a tutorial on background material and a workshop at research level, in addition to seminars and informal discussions.
IMS Membership is not required for participation in workshops or tutorials. For attendance at these activities, please complete the registration form (MSWord|PDF|PS) and fax it to us at (65) 6873 8292 or email it to us at ims@nus.edu.sg.
If you are an IMS member or are applying for IMS membership, you do not need to register for these activities.
The Institute for Mathematical Sciences invites applications for membership for participation in the above program. Limited funds to cover travel and living expenses are available to young scientists. Applications should be received at least six (6) months before the commencement of membership. Application form is available in MSWord | PDF | PS format for download.
For enquiries on scientific aspects of the program, please email Andrew Barbour at adb@amath.unizh.ch or Louis Chen at lhychen@ims.nus.edu.sg.
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