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Nearest neighbors and Voronoi volumes in high-dimensional point processes with various distance functions

Published online by Cambridge University Press:  01 July 2016

Charles M. Newman  and
Yosef Rinott
Charles M. Newman*
Affiliation:
University of Arizona
Yosef Rinott*
Affiliation:
Hebrew University of Jerusalem
*
∗Postal address: Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA.
∗∗Postal address: Department of Statistics, The Hebrew University of Jerusalem, Jerusalem 91905, Israel.
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Abstract

Consider a Poisson point process of density 1 in Rd, centered so that the origin is one of the points. Using lv distances, 1≦p≦∞, define Nd as the number of other points which have the origin as their nearest neighbor and Vol Vd as the volume of the Voronoi region of the origin. We prove that Nd → Poisson (λ = 1) and Vol Vd → 1 in distribution as d →∞, thus extending previous results from the case p = 2. More generally, for a variety of exchangeable distributions for n + 1 points, e0, · ··, en, in Rd and a variety of distances, we obtain the asymptotic behavior of Ndn, the number of points which have e0 as their nearest neighbor, as n, d → ∞ in one or both of the possible iterated orders. The distributions treated include points distributed on the unit l2 sphere and the distances treated include non-lp distances related to correlation coefficients.

Keywords

POISSON PROCESSESCONVERGENCE IN DISTRIBUTIONCORRELATIONAL DISTANCES
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 4 , December 1985 , pp. 794 - 809
Copyright
Copyright © Applied Probability Trust 1985 

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Footnotes

Research supported in part by NSF Grant MCS 80–19384 and by a Lady Davis visiting professorship at the Hebrew University of Jerusalem.

Research supported by grant 3265/83 from the United States-Israel Binational Science Foundation.

References

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