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Nearest neighbors and Voronoi regions in certain point processes

Published online by Cambridge University Press:  01 July 2016

C. M. Newman ,
Y. Rinott  and
A. Tversky
C. M. Newman*
Affiliation:
University of Arizona
Y. Rinott*
Affiliation:
The Hebrew University of Jerusalem
A. Tversky*
Affiliation:
Stanford University
*
∗Alfred P. Sloan Research Fellow, Department of Mathematics, University of Arizona, Tucson, AZ 85721, U.S.A.
∗∗Postal address: Department of Statistics, Faculty of Social Sciences, The Hebrew University of Jerusalem, Jerusalem, Israel.
∗∗∗Postal address: Department of Psychology, Stanford University, Stanford, CA 94305, U.S.A.
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Abstract

We investigate, for several models of point processes, the (random) number N of points which have a given point as their nearest neighbor. The largedimensional limit of Poisson processes is treated by considering for n points independently and uniformly distributed in a d-dimensional cube of volume n and showing that Poisson (λ= 1). An asymptotic Poisson (λ= 1) distribution also holds for many of the other models. On the other hand, we find that . Related results concern the (random) volume, , of a Voronoi polytope (or Dirichlet cell) in the cube model; we find that while

Keywords

POISSON PROCESSESDIRICHLET CELLS
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 4 , December 1983 , pp. 726 - 751
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

Research supported in part by NSF Grant MCS 80-19384.

Research supported in part by NSF Grant MCS 79-24310,A2 and NIH Grant 5R01-GM10452-19 at Stanford University.

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