# 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.