*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 ξ

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

**Case 2:** Majority of P(ξ_{i} ≠ 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:

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