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An analysis of the Pólya point process

Published online by Cambridge University Press:  01 July 2016

Ed Waymire  and
Vijay K. Gupta
Ed Waymire*
Affiliation:
Oregon State University
Vijay K. Gupta*
Affiliation:
University of Mississippi
*
∗Postal address: Department of Mathematics, Oregon State University, Corvallis, OR 97331, U.S.A.
∗∗Postal address: Department of Civil Engineering, University of Mississippi, MS 38677, U.S.A.
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Abstract

The Pólya process is employed to illustrate certain features of the structure of infinitely divisible stochastic point processes in connection with the representation for the probability generating functional introduced by Milne and Westcott in 1972. The Pólya process is used to provide a counterexample to the result of Ammann and Thall which states that the class of stochastic point processes with the Milne and Westcott representation is the class of regular infinitely divisble point processes. So the general representation problem is still unsolved. By carrying the analysis of the Pólya process further it is possible to see the extent to which the general representation is valid. In fact it is shown in the case of the Pólya process that there is a critical value of a parameter above which the representation breaks down. This leads to a proper version of the representation in the case of regular infinitely divisible point processes.

Keywords

STOCHASTIC POINT PROCESSREGULAR AND SINGULAR INFINITELY DIVISIBLE STOCHASTIC POINT PROCESSPROBABILITY GENERATING FUNCTIONALGAUSS-POISSON PROCESSPÓLYA PROCESS

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 1 , March 1983 , pp. 39 - 53
Copyright
Copyright © Applied Probability Trust 1983 

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