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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.

References

Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Cox, T. F. (1981) Reflexive nearest neighbors. Biometrics 37, 367–369.Google Scholar
Esary, J. D., Proschan, F. and Walkup, D. W. (1967) Association of random variables with applications. Ann. Math. Statist. 38, 1466–1474.Google Scholar
Gilbert, E. N. (1962) Random subdivisions of space into crystals. Ann. Math. Statist. 33, 958–972.Google Scholar
Gnedenko, B. V. and Kolmogorov, A. N. (1954) Limit Distributions For Sums of Independent Random Variables. Addison-Wesley, Cambridge, Mass.Google Scholar
Karlin, S. and Taylor, H. M. (1981) A Second Course In Stochastic Processes. Academic Press, New York.Google Scholar
Leech, J. and Sloane, N. J. A. (1971) Sphere packings and error-correcting codes. Canad. J. Math. 23, 718–745.CrossRefGoogle Scholar
Matheron, G. (1975) Random Sets and Integral Geometry. Wiley, London.Google Scholar
Miles, R. E. (1974) A synopsis of ‘Poisson Flats in Euclidean Spaces’. In Stochastic Geometry, ed. Harding, E. F. and Kendall, D. G., Wiley, London, 202–227.Google Scholar
Newman, C. M. (1980) Normal fluctuations and the FKG inequalities. Commun. Math. Phys. 74, 119–128.Google Scholar
Newman, C. M. and Wright, A. L. (1981) An invariance principle for certain dependent sequences. Ann. Prob. 9, 671–675.Google Scholar
Roberts, F. D. K. (1969) Nearest neighbours in a Poisson ensemble. Biometrika 56, 401–406.Google Scholar
Slepian, D. (1962) The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41, 463–501.Google Scholar
Schwarz, G. and Tversky, A. (1980) On the reciprocity of proximity relations. J. Math. Psychol. 22, 157–175.Google Scholar
Tversky, A., Rinott, Y. and Newman, C. M. (1984) Nearest neighbor analysis of point processes: applications to multidimensional scaling. J. Math. Psychol. To appear.CrossRefGoogle Scholar
Wood, T. E. (1982) Sequences of Associated Random Variables. Ph.D. Dissertation, University of Virginia.Google Scholar