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On interpoint distances for planar Poisson cluster processes

Published online by Cambridge University Press:  14 July 2016

Richard J. Kryscio  and
Roy Saunders
Richard J. Kryscio*
Affiliation:
University of Kentucky, Lexington
Roy Saunders*
Affiliation:
American Critical Care
*
∗ Postal address: Department of Statistics, University of Kentucky, Lexington, KY 40506, U.S.A. Research carried out while the author was on leave from Northern Illinois University.
∗∗ Postal address: American Critical Care, 1600 Waukegan Road, McGaw Park, IL 60085, U.S.A.
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Abstract

For stationary Poisson or Poisson cluster processes ξ on R2 we study the distribution of the interpoint distances using the interpoint distance function and the nearest-neighbor indicator function . Here Sr (x) is the interior of a circle of radius r having center x, I(t) is that subset of D which has x ∊ D and St(x) ⊂ D and χ is the usual indicator function. We show that if the region D ⊂ R2 is large, then these functions are approximately distributed as Poisson processes indexed by and , where µ(D) is the Lebesgue measure of D.

Keywords

POINT PROCESSNEAREST-NEIGHBOUR DISTANCEWEAK CONVERGENCETESTS FOR SPATIAL RANDOMNESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 3 , September 1983 , pp. 513 - 528
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

Work supported by the National Science Foundation (MCS-79–03781) while both authors were employed by the Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115.

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