Tid: 15 november 2004 kl 1515-1700

Plats : Seminarierummet 3733, Institutionen för matematik, KTH, Lindstedts väg 25, plan 7. Karta!

Föredragshållare: Professor Aihua Xia, Department of Mathematics and Statistics, University of Melbourne.

Titel: Stein's method: from Poisson approximation to a discrete central limit theorem

Sammanfattning: We want to approximate P(W ε A) for a set A in {...,-2,-1,0,1,2,...} and W a sum of independent (or weakly dependent) integer-valued random variables ξ12,...,ξn with finite second moments. There are two cases to consider.

Case 1: Majority of P(ξi≠ 0), i=1,...,n, are small, for example, W counts the number of occurrences of certain rare events.

Case 2: Majority of P(ξi ≠ 0), i=1,...,n, are relatively large, then the distribution of W should behave like a "discrete normal".

The first case is well approximated by a Poisson or a modified Poisson such as compound Poisson, Poisson signed measures with errors of approximation estimated by Stein-Chen method. Barbour's probabilistic interpretation of Stein-Chen method for estimating the error of Poisson approximation not only paved a way for investigating Poisson process approximation, but also provided an opportunity for studying other approximations. Xia (1999) gave a purely probabilistic proof of Stein bound for Poisson approximation, and the case of approximations by general distributions on the non-negative integers was studied by Brown and Xia (2001). The methods in Brown and Xia (2001) apply to a very large class of approximating distributions on the non-negative integers, including Poisson, binomial, negative binomial, as well as a natural class for higher-order approximations by probability distributions rather than signed measures. This offers a comprehensive solution to case 1.

In terms of case 2, Goldstein and Xia found a family of discrete distributions which behave in the same way as normal does in the central limit theorem. This talk will cover the following topics:

  1. The principle of Stein's method for discrete distribution approximations.
  2. Why do we need to have Markov birth-death processes in this exercise?
  3. Around Poisson approximation.
  4. Polynomial birth-death (PBD) approximation.
  5. Zero biasing and a discrete central limit theorem.

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