Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T00:13:09.168Z Has data issue: false hasContentIssue false
  • English
  • Français

A limit theorem for spatial point processes

Published online by Cambridge University Press:  01 July 2016

Steven P. Ellis
Steven P. Ellis*
Affiliation:
Massachusetts Institute of Technology
*
∗Present address: Department of Statistics, University of Rochester, Rochester, NY 14627, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Spatial point processes are considered whose points are subjected to certain classes of affine transformations indexed by some variable, T. Under some hypotheses, for large T integrals with respect to such a point process behave approximately as if the process were Poisson. Under stronger hypotheses, the transformed process converges as a process to a Poisson process. The result gives the asymptotic distribution of certain density estimates.

Keywords

POISSON LIMITDENSITY ESTIMATION
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 3 , September 1986 , pp. 646 - 659
Copyright
Copyright © Applied Probability Trust 1986 

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.)

Footnotes

This work was partially supported by United States National Science Foundation Grants MCS 75-10376, PFR 79-01642, MCS 82-01732, and MCS 82-02122

References

Bernstein, S. (1927) Sur l&extension du théorème limite du calcul des probabilités aux sommes de quantités dépendantes. Math. Ann. 97, 1–59.Google Scholar
Brillinger, D. R. (1976) Estimation of the second-order intensities of a bivariate stationary point process. J. R. Statist. Soc. B 38, 60–66.Google 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, 148–161.Google Scholar
Ellis, S. P. (1981) Density Estimation for Point Process Data. Ph.D. Dissertation, Department of Statistics, University of California, Berkeley.Google Scholar
Ellis, S. P. (1983a) Density estimation for multivariate data generated by a point process. Massachusetts Institute of Technology Statistics Center, NSF Report #39.Google Scholar
Ellis, S. P. (1983b) Density estimation for point processes. Submitted to Ann. Statist. Google Scholar
Ellis, S. P. (1983C) Second order approximation to the characteristic functions of certain point process integrals. Submitted to Adv. Appl. Prob. Google Scholar
Isham, V. (1980) Dependent thinning of point processes. J. Appl. Prob. 17, 987–995.CrossRefGoogle Scholar
Jagers, P. and Lindvall, T. (1974) Thinning and rare events in point processes. Z. Wahrscheinlichkeitsth. 28, 89–98.CrossRefGoogle Scholar
Kallenberg, O. (1976) Random Measures. Akademie-Verlag, Berlin; Academic Press, New York.Google Scholar
Simmons, G. F. (1963) Introduction to Topology and Modern Analysis. McGraw-Hill, New York.Google Scholar
Volkonskii, V. A. and Rozanov, Yu. A. (1959) Some limit theorems for random functions. I. Theory Prob. Appl. 4, 178–197.Google Scholar