# New versions of Berry--Esseen theorems

*Stochastic Processes Seminar Series*

*by Aihua Xia*

*Institution:*Department of Mathematics and Statistics, The University of Melbourne

*Date: Thu 26th May 2005*

*Time: 1:15 PM*

*Location: Room 213, Richard Berry Building*

*Abstract*: The talk will focus on two topics that I finished during my SSP(L) leave.

The first (joint work with A. D. Barbour) is on the accuracy, in terms of

the Kolmogorov-Smirnoff distance, of normal approximation to random

variables resulting from integrating a random field with respect to a

point process. We illustrate the use of our Berry--Esseen theorems in two

examples: a rather general model of the insurance collective and in

counting the maximal points in a 2-dimensional region.

The second part is a discrete version of the central limit theorem and is

based on a joint work with L. Goldstein. Here, we consider the sum $W$ of

independent (or weakly dependent) integer-valued random variables

$\xi_1$, $\xi_2$, ..., $\xi_n$ with finite second moments. The actual probabilities of events

to do with $W$ are often impossible to calculate and our interest is to

find a suitable distribution which is easily computable and approximates

the distribution of $W$ reasonably well (i.e., the approximation error in

terms the total variation distance is reasonably small). Using a concept

called zero biasing and Stein's method, we find a family of discrete

distributions that behave in the

same way as normal distribution does in the central limit theorem. We

establish a Berry--Esseen theorem for the discrete version of the central

limit theorem.

*For More Information:* Aihua Xia email: a.xia@ms.unimelb.edu.au