Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-16T09:24:30.231Z Has data issue: false hasContentIssue false
  • English
  • Français

Count distributions, orderliness and invariance of Poisson cluster processes

Published online by Cambridge University Press:  14 July 2016

Larry P. Ammann  and
Peter F. Thall
Larry P. Ammann*
Affiliation:
A. D
Peter F. Thall*
Affiliation:
University of Texas at Dallas
*
∗Postal address: Programs in Mathematical Sciences, University of Texas at Dallas, Box 688, Richardson, TX 75080, U.S.A.
∗Postal address: Programs in Mathematical Sciences, University of Texas at Dallas, Box 688, Richardson, TX 75080, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

The probability generating functional (p.g.fl.) of a non-homogeneous Poisson cluster process is characterized in Ammann and Thall (1977) via a decomposition of the KLM measure of the process. This p.g.fl. representation is utilized in the present article to show that the family 𝒟 of Poisson cluster processes with a.s. finite clusters is invariant under a class of cluster transformations. Explicit expressions for the finite-dimensional count distributions, product moment measures, and the distribution of clusters are derived in terms of the KLM measure. It is also shown that an element of 𝒟 has no multiple events iff the points of each cluster are a.s. distinct.

Keywords

POINT PROCESSPOISSON CLUSTER PROCESSPROBABILITY GENERATING FUNCTIONALORDERLINESSFINITE-DIMENSIONAL DISTRIBUTIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 16 , Issue 2 , June 1979 , pp. 261 - 273
Copyright
Copyright © Applied Probability Trust 1979 

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 aftereffects. Ann. Prob. 6, 216–230.Google Scholar
Brown, M. (1969) An invariance property of Poisson processes. J. Appl. Prob. 6, 453–458.CrossRefGoogle Scholar
Daley, D. J. (1974) Various concepts of orderliness for point processes. In Stochastic Geometry, ed. Harding, E. F. and Kendall, D. G. Wiley, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (1972) A summary of the theory of point processes. In Stochastic Point Processes: Statistical Analysis, Theory and Applications, ed. Lewis, P.A.W., Wiley, New York.Google Scholar
Jagers, P. (1972) On the weak convergence of superpositions of point processes. Z. Wahrscheinlichkeitsth. 22, 1–7.Google Scholar
Jagers, P. (1974) Aspects of random measures and point processes. In Advances in Probability and Related Topics 3, ed. Ney, P. and Port, S. C. Dekker, New York, 179–239.Google Scholar
Kallenberg, O. (1973) Characterization and convergence of random measures and point processes. Z. Wahrscheinlichkeitsth 27, 9–21.Google Scholar
Kerstan, J. and Matthes, K. (1964) Stationary stochastic point processes II (in German). Jber. Deutsch Math-Verein. 66, 106–118. English translation (1971) University of North Carolina Mimeo Series 775.Google Scholar
Kerstan, J., Matthes, K. and Mecke, J. (1974) Unbergrenzt teilbare Punktprozesse. Akademie-Verlag, Berlin.Google Scholar
Kingman, J. F. C. (1967) Completely random measures. Pacific J. Math. 21, 59–78.CrossRefGoogle Scholar
Lee, P. M. (1967) Infinitely divisible stochastic processes. Z. Wahrscheinlichkeitsth. 7, 147–160.Google Scholar
Matthes, K. (1963) Stationäre zufällige Punktfolgen, I. Jber. Deutsch Math-Verein 66, 66–79.Google Scholar
Matthes, K. (1972) Infinitely divisible point processes. In Stochastic Point Processes: Statistical Analysis, Theory and Applications, ed. Lewis, P. A. W., 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.Google Scholar
Moyal, J. E. (1962) The general theory of stochastic population processes. Acta Math. 108, 1–31.Google 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.CrossRefGoogle Scholar
Westcott, M. (1972) The probability generating functional. J. Austral. Math. Soc. 14, 448–466.CrossRefGoogle Scholar