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On interpoint distances for planar Poisson cluster processes
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- Journal:
- Journal of Applied Probability / Volume 20 / Issue 3 / September 1983
- Published online by Cambridge University Press:
- 14 July 2016, pp. 513-528
- Print publication:
- September 1983
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- Article
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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.
and the nearest-neighbor indicator function 
. Here 
and 
, where 