Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-02T23:37:12.921Z Has data issue: false hasContentIssue false
  • English
  • Français

Some limit theorems for clustered occupancy models

Published online by Cambridge University Press:  14 July 2016

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

Abstract

Most generalizations of the classical occupancy model involve non-homogeneous shot assignment probabilities, but retain the independence of the individual shot assignments. Hence, these models are associated with non-homogeneous Poisson processes. The present article discusses a generalization in which the shot assignments are not independent, but which result in clustering of the shots. Conditions are given under which this clustered occupancy model converges to a Poisson cluster process. Limiting distributions for the number of empty cells are obtained for various allocation intensities when the total number of shots is deterministic as well as random. In particular, it is shown that when the allocation is sparse, then the limiting distribution of the number of empty cells is compound Poisson.

Keywords

POISSON CLUSTER PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 4 , December 1983 , pp. 788 - 802
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. (1981) Occupancy model approximations of Poisson cluster processes. The University of Texas at Dallas Technical Report #99.Google Scholar
Ammann, L. P. and Thall, P. F. (1977) On the structure of regular infinitely divisible point processes. Stoch. Proc. Appl. 6, 87–94.Google Scholar
Ammann, L. P. and Thall, P. F. (1979) Count distributions, orderliness, and invariance of Poisson cluster processes. J. Appl. Prob. 16, 261–273.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, ed. Lewis, P. A. W., Wiley, New York, 299–383.Google Scholar
Johnson, N. L. and Kotz, S. (1977) Urn Models and Their Applications. Wiley, New York.Google Scholar
Kallenberg, O. (1976) Random Measures. Akademie-Verlag, Berlin.Google Scholar
Kolchin, V. F., Sevast'yanov, B. A. and Chistyakov, V. P. (1978) Random Allocations. Winston, Washington, D.C. Google Scholar
Loéve, M. (1963) Probability Theory, 3rd edn. Van Nostrand Reinhold, New York.Google Scholar