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Poisson approximation in connection with clustering of random points. (English) Zbl 0941.60027

Let \(n\) particles be independently and uniformly distributed in a rectangle \({\mathbf A}\subset\mathbb{R}\). The number of \(k\) subsets of particles which are actually covered by some translate of a given convex set \({\mathbf A}\subset{\mathbf C}\) is denoted by \(W\). The positions of such subsets constitute a point process on \({\mathbf A}\). 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 paper considers Poisson (process) approximations of \(W\) and of the above point process, by way of Stein’s method.

MSC:

60D05 Geometric probability and stochastic geometry
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F17 Functional limit theorems; invariance principles
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[1] ALDOUS, D. 1989. Probability Approximation Via the Poisson Clumping Heuristic. Springer, New York. Z. · Zbl 0679.60013
[2] 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. · Zbl 0518.60062
[3] ALM, S. E. 1997. On the distribution of scan statistics of a two-dimensional Poisson process. Adv. in Appl. Probab. 29 1 18. Z. JSTOR: · Zbl 0883.60046
[4] ARRATIA, R., GOLDSTEIN, L. and GORDON, L. 1989. Two moments suffice for Poisson approximations: the Chen Stein method. Ann. Probab. 17 9 25. Z. · Zbl 0675.60017
[5] 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. JSTOR: · Zbl 0511.60025
[6] BARBOUR, A. D. and EAGLESON, G. K. 1984. Poisson convergence for dissociated statistics. J. Roy. Statist. Soc. Ser. B 46 397 402. Z. JSTOR: · Zbl 0567.62062
[7] BARBOUR, A. D., HOLST, L. and JANSON, S. 1992. Poisson Approximation. Oxford Univ. Press. Z. · Zbl 0765.60015
[8] BERWALD, W. and VARGA, O. 1937. Integralgeometrie 24, uber die Schiebungen im Raum. Math. \" Z. 42 710 736. · JFM 63.0677.03
[9] BLASCHKE, W. 1937. Integralgeometrie 21, uber Schiebungen. Math. Z. 42 399 410. \" Z. · JFM 63.0677.02
[10] BONNESEN, T. and FENCHEL, W. 1948. Theorie der Konvexen Korper. Chelsea, New York. \" Z. · Zbl 0008.07708
[11] CHEN, L. H. Y. 1975. Poisson approximation for dependent trials. Ann. Probab. 3 534 545. Z. · Zbl 0335.60016
[12] 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. · Zbl 0063.01222
[13] GLAZ, J. 1989. Approximation and bounds for the distribution of the scan statistic. J. Amer. Statist. Assoc. 84 560 566. Z. JSTOR: · Zbl 0677.62015
[14] JANSON, S. 1984. Bounds on the distributions of extremal values of a scanning process. Stochastic Proces. Appl. 18 313 328. Z. · Zbl 0549.60066
[15] KRy SCIO, R. J. and SAUNDERS, R. 1983. On interpoint distances for planar Poisson cluster processes, J. Appl. Probab. 20 513 528. Z. JSTOR: · Zbl 0527.60049
[16] LOADER, C. R. 1991. Large-deviation approximations to the distribution of scan statistics. Adv. Appl. Probab. 23 751 771. Z. Z. JSTOR: · Zbl 0741.60036
[17] 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. · Zbl 0031.06003
[18] MACK, C. 1949. The expected number of aggregates in a random distribution of n points. Proc. Cambridge Philos. Soc. 46 285 292. Z. · Zbl 0036.20901
[19] MANSSON, M. 1996. On clustering of random points in the plane and in space. Thesis, Dept. Mathematics, Chalmers Univ. Technology. Z.
[20] 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. · Zbl 0313.53026
[21] MCGINLEY, W. G. and SIBSON, R. 1975. Dissociated random variables. Math. Proc. Cambridge Philos. Soc. 77 185 188. Z. · Zbl 0353.60018
[22] NAUS, J. I. 1982. Approximations for distributions of scan statistics. J. Amer. Statist. Assoc. 77 177 183. Z. JSTOR: · Zbl 0482.62010
[23] SILBERSTEIN, L. 1945. The probable number of aggregates in random distributions of points. Philos. Mag. 36 319 336. Z.
[24] SILVERMAN, B. and BROWN, T. 1978. Short distances, flat triangles and Poisson limits. J. Appl. Probab. 15 815 825. Z. JSTOR: · Zbl 0396.60029
[25] SILVERMAN, B. and BROWN, T. 1979. Rates of Poisson convergence for U-statistics. J. Appl. Probab. 16 428 432. Z. JSTOR: · Zbl 0446.62020
[26] WEIL, W. 1990. Iterations of translative formulae and non-isotropic Poisson processes of particles. Math. Z. 205 531 549. · Zbl 0693.52001
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