Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-16T13:23:27.964Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ammann, L. P. and Thall, P. F. (1977) On the structure of regular infinitely divisible point processes. Stoch. Proc. Appl. 6, 87–94.CrossRefGoogle Scholar
Ammann, L. P. and Thall, P. F. (1978) Random measures with after effects. Ann. Prob. 6, 216–230.CrossRefGoogle Scholar
Brillinger, D. R. (1978) A note on a representation for the Gauss-Poisson process. Stoch. Proc. Appl. 6, 135–137.Google Scholar
Daley, D. J. and Vere-Jones, D. (1972) A summary of the theory of point processes. In Stochastic Point Processes, ed. Lewis, P. A. W. Wiley, New York.Google Scholar
Dunford, N. and Schwartz, J. T. (1958) Linear Operators, Part I. Interscience, New York.Google Scholar
Grandell, J. (1976) Doubly Stochastic Poisson Processes. Lecture Notes in Mathematics 529, Springer-Verlag, New York.Google Scholar
Lévy, P. (1937) Sur les exponentielles de polynomes. Ann. Sci. Ecole Norm. Sup. 54, 231–292.Google Scholar
Macchi, O. (1975) The coincidence approach to stochastic point processes. Adv. Appl. Prob. 7, 83–122.Google Scholar
Matthes, K., Kerstan, J. and Mecke, J. (1978) Infinitely Divisible Point Processes. Wiley, New York.Google Scholar
Milne, R. K. and Westcott, M. (1972) Further results for the Gauss-Poisson process. Adv. Appl. Prob. 4, 151–176.CrossRefGoogle Scholar
Newman, D. S. (1970) A new family of point processes which are characterized by their second moment properties. J. Appl. Prob. 7, 338–358.Google Scholar
Waymire, E. and Gupta, V. K. (1981) The mathematical structure of rainfall representations. Water Resources Res. 17, 1261–1294.CrossRefGoogle Scholar
Westcott, M. (1972) The probability generating functional. J. Austral. Math. Soc. 14, 448–466.Google Scholar