The Annals of Applied Probability

A large-dimensional independent and identically distributed property for nearest neighbor counts in Poisson processes

Gordon Simons and Yi-Ching Yao

Full-text: Open access


For an arbitrary point of a homogeneous Poisson point process in a d-dimensional Euclidean space, consider the number of Poisson points that have that given point as their rth nearest neighbor $(r = 1, 2, \dots)$. It is shown that as d tends to infinity, these nearest neighbor counts $(r = 1, 2, \dots)$ are iid asymptotically Poisson with mean 1. The proof relies on Rényi's characterization of Poisson processes and a representation in the limit of each nearest neighbor count as a sum of countably many dependent Bernoulli random variables.

Article information

Ann. Appl. Probab., Volume 6, Number 2 (1996), 561-571.

First available in Project Euclid: 18 October 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G55: Point processes
Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Nearest neighbor counts Poisson point process


Yao, Yi-Ching; Simons, Gordon. A large-dimensional independent and identically distributed property for nearest neighbor counts in Poisson processes. Ann. Appl. Probab. 6 (1996), no. 2, 561--571. doi:10.1214/aoap/1034968144.

Export citation


  • COX, T. F. 1981. Reflexive nearest neighbors. Biometrics 37 367 369. Z.
  • HALMOS, P. R. 1950. Measure Theory. Van Nostrand, New York. Z.
  • HENZE, N. 1986. On the probability that a random point is the jth nearest neighbor to its own kth nearest neighbor. J. Appl. Probab. 23 221 226. Z.
  • HENZE, N. 1987. On the fraction of random points with specified nearest-neighbor interrelations and degree of attraction. Adv. in Appl. Probab. 19 873 895. Z.
  • KINGMAN, J. F. C. 1993. Poisson Processes. Oxford Univ. Press. Z.
  • NEWMAN, C. M. and RINOTT, Y. 1985. Nearest neighbors and Voronoi volumes in high-dimensional point processes with various distance functions. Adv. in Appl. Probab. 17 794 809. Z.
  • NEWMAN, C. M., RINOTT, Y. and TVERSKY, A. 1983. Nearest neighbors and Voronoi regions in certain point processes. Adv. in Appl. Probab. 15 726 751. Z.
  • PICARD, D. K. 1982. Isolated nearest neighbors. J. Appl. Probab. 19 444 449. Z.
  • ROBERTS, F. D. K. 1969. Nearest neighbors in a Poisson ensemble. Biometrika 56 401 406. Z.
  • SCHILLING, M. F. 1986. Mutual and shared neighbor probabilities: finiteand infinite-dimensional results. Adv. in Appl. Probab. 18 388 405. Z.
  • SCHWARZ, G. and TVERSKY, A. 1980. On the reciprocity of proximity relations. J. Math. Psy chol. 22 157 175. Z.
  • TVERSKY, A., RINOTT, Y. and NEWMAN, C. M. 1983. Nearest neighbor analysis of point processes: applications to multidimensional scaling. J. Math. Psy chol. 27 235 250.