Advances in Applied Probability

On comparison of clustering properties of point processes

Bartłomiej Błaszczyszyn and D. Yogeshwaran

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In this paper, we propose a new comparison tool for spatial homogeneity of point processes, based on the joint examination of void probabilities and factorial moment measures. We prove that determinantal and permanental processes, as well as, more generally, negatively and positively associated point processes are comparable in this sense to the Poisson point process of the same mean measure. We provide some motivating results on percolation and coverage processes, and preview further ones on other stochastic geometric models, such as minimal spanning forests, Lilypond growth models, and random simplicial complexes, showing that the new tool is relevant for a systemic approach to the study of macroscopic properties of non-Poisson point processes. This new comparison is also implied by the directionally convex ordering of point processes, which has already been shown to be relevant to the comparison of the spatial homogeneity of point processes. For this latter ordering, using a notion of lattice perturbation, we provide a large monotone spectrum of comparable point processes, ranging from periodic grids to Cox processes, and encompassing Poisson point processes as well. They are intended to serve as a platform for further theoretical and numerical studies of clustering, as well as simple models of random point patterns to be used in applications where neither complete regularity nor the total independence property are realistic assumptions.

Article information

Adv. in Appl. Probab., Volume 46, Number 1 (2014), 1-20.

First available in Project Euclid: 1 April 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G55: Point processes 60E15: Inequalities; stochastic orderings
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G60: Random fields

Point process clustering directionally convex ordering association perturbed lattice determinantal point process permanental point process sub-Poisson point process super-Poisson point process


Błaszczyszyn, Bartłomiej; Yogeshwaran, D. On comparison of clustering properties of point processes. Adv. in Appl. Probab. 46 (2014), no. 1, 1--20. doi:10.1239/aap/1396360100.

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  • Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation (Oxford Studies Prob. 2). Oxford University Press.
  • Bodini, O., Gardy, D. and Roussel, O. (2012). Boys-and-girls birthdays and Hadamard products. Fund. Inform. 117, 85–104.
  • Boucheron, S. and Gardy, D. (1997). An urn model from learning theory. Random Structures Algorithms 10, 43–67.
  • Borcea, J., Brändén, P. and Liggett, T. M. (2009). Negative dependence and the geometry of polynomials. J. Amer. Math. Soc. 22, 521–567.
  • Dubhashi, D. and Panconesi, A. (2009). Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press.
  • Durrett, R. (1996). Probability: Theory and Examples, 2nd edn. Duxbury Press, Belmont, CA.
  • Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Application. John Wiley, New York.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edn. John Wiley, New York.
  • Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge University Press.
  • Gnedin, A., Hansen, B. and Pitman, J. (2007). Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws. Prob. Surveys 4, 146–171.
  • Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables, with applications. Ann. Statist. 11, 286–295.
  • Kolchin, V. F., Sevast'yanov, B. A. and Chistyakov, V. P. (1978). Random Allocations. V. H. Winston, Washington, DC.
  • Nakata, T. (2008). A Poisson approximation for an occupancy problem with collisions. J. Appl. Prob. 45, 430–439.
  • Nakata, T. (2008). Collision probability for an occupancy problem. Statist. Prob. Lett. 78, 1929–1932.
  • Nishimura, K. and Sibuya, M. (1988). Occupancy with two types of balls. Ann. Inst. Statist. Math. 40, 77–91.
  • Pemantle, R. (2000). Towards a theory of negative dependence. J. Math. Phys. 41, 1371–1390.
  • Popova, T. Yu. (1968). Limit theorems in a model of distribution of particles of two types. Theory Prob. Appl. 13, 511–516.
  • Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
  • Selivanov, B. I. (1995). On the waiting time in a scheme for the random allocation of colored particles. Discrete Math. Appl. 5, 73–82.
  • Wendl, M. C. (2003). Collision probability between sets of random variables. Statist. Prob. Lett. 64, 249–254.
  • Wendl, M. C. (2005). Probabilistic assessment of clone overlaps in DNA fingerprint mapping via a priori models. J. Comput. Biol. 12, 283–297.