PSU Mathematics Department - Meeting Details

For more information about this meeting, contact Anatole Katok, Svetlana Katok.

Abstract:
Title:	Two theorems of de Moivre and Poisson: Recent extensions with applications: ATTENTION THIS COLLOQUIUM TALK WILL START AT 3:30pm.
Seminar:	Department of Mathematics Colloquium
Speaker:	Manfred Denker, University of Gottingen and Case Western Reserve University
One of the oldest results in probability is de Moivre's theorem on the approximation of binomial probability functions by normal densities. It dates from 1733 and is published in his {\it Doctrine of Chance}. It may be seen as the first results on local limit theorems. Another (simple) result is Poisson's approximation of binomial probability functions by a Poisson distribution from 1837. The talk will discuss some extensions of each of the results. For the first one we demonstrate the use of Richter's multidimensional local limit theorem to analyse an extension of the Cox-Ross-Rubinstein model for European call options, deriving a sharper form of the Black-Scholes formula. A further extension of local limit theory has recently been developed in form of almost sure versions, which is based on the scaling property of Brownian motion. The extension of the second old result is in fact based on Sevastyanov's theorem on Poisson approximation of binary responses. We give the extension to the multinomial case (this is an occupancy problem) and discuss an application to U-statistics, extending a result by Brown and Silverman.