The Annals of Applied Probability

Poisson approximation in connection with clustering of random points

Marianne Månsson

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Let n particles be independently and uniformly distributed in a rectangle $\mathbf{A} \subset \mathbb{R}^2$. Each subset consisting of $k \leq n$ particles may possibly aggregate in such a way that it is covered by some translate of a given convex set $C \subset \mathbf{A}$. The number of k-subsets which actually are covered by translates of C is denoted by W. The positions of such subsets constitute a point process on A. Each point of this process can be marked with the smallest necessary "size" of a set, of the same shape and orientation as C, which covers the particles determining the point. This results in a marked point process.

The purpose of this paper is to consider Poisson process approximations of W and of the above point processes, by means of Stein's method. To this end, the exact probability for k specific particles to be covered by some translate of C is given.

Article information

Ann. Appl. Probab., Volume 9, Number 2 (1999), 465-492.

First available in Project Euclid: 21 August 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 52A22: Random convex sets and integral geometry [See also 53C65, 60D05] 60G55: Point processes

Poisson approximation Stein's method total variation distance integral geometry convex sets mixed areas Poisson process


Månsson, Marianne. Poisson approximation in connection with clustering of random points. Ann. Appl. Probab. 9 (1999), no. 2, 465--492. doi:10.1214/aoap/1029962751.

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  • ALDOUS, D. 1989. Probability Approximation Via the Poisson Clumping Heuristic. Springer, New York. Z.
  • ALM, S. E. 1983. On the distribution of the scan statistic of a Poisson process. In Probability Z and Mathematical Statistics. Essay s in Honour of Carl-Gustav Esseen 1 10. A. Gut. and L. Holst, eds., Dept. Mathematics, Uppsala Univ. Z.
  • ALM, S. E. 1997. On the distribution of scan statistics of a two-dimensional Poisson process. Adv. in Appl. Probab. 29 1 18. Z.
  • ARRATIA, R., GOLDSTEIN, L. and GORDON, L. 1989. Two moments suffice for Poisson approximations: the Chen Stein method. Ann. Probab. 17 9 25. Z.
  • BARBOUR, A. D. and EAGLESON, G. K. 1983. Poisson approximation for some statistics based on exchangeable trials. Adv. in Appl. Probab. 15 585 600. Z.
  • BARBOUR, A. D. and EAGLESON, G. K. 1984. Poisson convergence for dissociated statistics. J. Roy. Statist. Soc. Ser. B 46 397 402. Z.
  • BARBOUR, A. D., HOLST, L. and JANSON, S. 1992. Poisson Approximation. Oxford Univ. Press. Z.
  • BERWALD, W. and VARGA, O. 1937. Integralgeometrie 24, uber die Schiebungen im Raum. Math. ¨ Z. 42 710 736.
  • BLASCHKE, W. 1937. Integralgeometrie 21, uber Schiebungen. Math. Z. 42 399 410. ¨ Z.
  • BONNESEN, T. and FENCHEL, W. 1948. Theorie der Konvexen Korper. Chelsea, New York. ¨ Z.
  • CHEN, L. H. Y. 1975. Poisson approximation for dependent trials. Ann. Probab. 3 534 545. Z.
  • EGGLETON, P. and KERMACK, W. O. 1944. A problem in the random distribution of particles, Proc. Roy. Soc. Edinburgh Sec. A 62 103 115. Z.
  • GLAZ, J. 1989. Approximation and bounds for the distribution of the scan statistic. J. Amer. Statist. Assoc. 84 560 566. Z.
  • JANSON, S. 1984. Bounds on the distributions of extremal values of a scanning process. Stochastic Proces. Appl. 18 313 328. Z.
  • KRy SCIO, R. J. and SAUNDERS, R. 1983. On interpoint distances for planar Poisson cluster processes, J. Appl. Probab. 20 513 528. Z.
  • LOADER, C. R. 1991. Large-deviation approximations to the distribution of scan statistics. Adv. Appl. Probab. 23 751 771. Z. Z.
  • MACK, C. 1948. An exact formula for Q n, the probable number of k-aggregates in a random k distribution of n points. Philos. Mag. 39 778 790. Z.
  • MACK, C. 1949. The expected number of aggregates in a random distribution of n points. Proc. Cambridge Philos. Soc. 46 285 292. Z.
  • MANSSON, M. 1996. On clustering of random points in the plane and in space. Thesis, Dept. Mathematics, Chalmers Univ. Technology. Z.
  • MILES, R. E. 1974. The fundamental formula of Blaschke in integral geometry and geometrical probability and its iteration, for domains with fixed orientations. Austral. J. Statist. 16 111 118. Z.
  • MCGINLEY, W. G. and SIBSON, R. 1975. Dissociated random variables. Math. Proc. Cambridge Philos. Soc. 77 185 188. Z.
  • NAUS, J. I. 1982. Approximations for distributions of scan statistics. J. Amer. Statist. Assoc. 77 177 183. Z.
  • SILBERSTEIN, L. 1945. The probable number of aggregates in random distributions of points. Philos. Mag. 36 319 336. Z.
  • SILVERMAN, B. and BROWN, T. 1978. Short distances, flat triangles and Poisson limits. J. Appl. Probab. 15 815 825. Z.
  • SILVERMAN, B. and BROWN, T. 1979. Rates of Poisson convergence for U-statistics. J. Appl. Probab. 16 428 432. Z.
  • WEIL, W. 1990. Iterations of translative formulae and non-isotropic Poisson processes of particles. Math. Z. 205 531 549.