Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points
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Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points

T. Schreiber and J. E. Yukich

Source: Ann. Probab. Volume 36, Number 1 (2008), 363-396.

Abstract

We show that the random point measures induced by vertices in the convex hull of a Poisson sample on the unit ball, when properly scaled and centered, converge to those of a mean zero Gaussian field. We establish limiting variance and covariance asymptotics in terms of the density of the Poisson sample. Similar results hold for the point measures induced by the maximal points in a Poisson sample. The approach involves introducing a generalized spatial birth growth process allowing for cell overlap.

Primary Subjects: 60F05
Secondary Subjects: 60D05
Keywords: Convex hulls; maximal points; spatial birth growth processes; Gaussian limits

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Permanent link to this document: http://projecteuclid.org/euclid.aop/1196268683
Digital Object Identifier: doi:10.1214/009117907000000259

References

Bai, Z.-D., Chao, C.-C., Hwang, H.-K. and Liang, W.-Q. (1998). On the variance of the number of maxima in random vectors and applications. Ann. Appl. Probab. 8 886--895.
Mathematical Reviews (MathSciNet): MR1627803
Digital Object Identifier: doi:10.1214/aoap/1028903455
Project Euclid: euclid.aoap/1028903455
Bai, Z.-D., Hwang, H.-K., Liang, W.-Q. and Tsai, T.-H. (2001). Limit theorems for the number of maxima in random samples from planar regions. Electronic J. Probab. 6 1--41.
Mathematical Reviews (MathSciNet): MR1816046
Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation. Oxford Univ. Press.
Mathematical Reviews (MathSciNet): MR1163825
Barbour, A. D. and Xia, A. (2001). The number of two dimensional maxima. Adv. in Appl. Probab. 33 727--750.
Mathematical Reviews (MathSciNet): MR1875775
Digital Object Identifier: doi:10.1239/aap/1011994025
Project Euclid: euclid.aap/1011994025
Barndorff-Nielsen, O. and Sobel, M. (1966). On the distribution of the number of admissible points in a vector. Theory Probab. Appl. 11 249--269.
Mathematical Reviews (MathSciNet): MR0207003
Baryshnikov, Yu. (2000). Supporting-points processes and some of their applications. Probab. Theory Related Fields 117 163--182.
Mathematical Reviews (MathSciNet): MR1771659
Digital Object Identifier: doi:10.1007/s004400050002
Baryshnikov, Yu. and Yukich, J. E. (2005). Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15 213--253.
Mathematical Reviews (MathSciNet): MR2115042
Digital Object Identifier: doi:10.1214/105051604000000594
Project Euclid: euclid.aoap/1106922327
Baryshnikov, Yu. and Yukich, J. E. (2005). Gaussian limits and maximal points. Preprint. Available at http://www.lehigh.edu/~jey0/publications.html.
Mathematical Reviews (MathSciNet): MR2115042
Digital Object Identifier: doi:10.1214/105051604000000594
Project Euclid: euclid.aoap/1106922327
Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
Mathematical Reviews (MathSciNet): MR0233396
Zentralblatt MATH: 0172.21201
Cabo, A. J. and Groeneboom, P. (1994). Limit theorems for functionals of convex hulls. Probab. Theory Related Fields 100 31--55.
Mathematical Reviews (MathSciNet): MR1292189
Digital Object Identifier: doi:10.1007/BF01204952
Chen, W.-M., Hwang, H.-K. and Tsai, T.-H. (2003). Efficient maxima finding algorithms for random planar samples. Discrete Math. Theor. Comput. Sci. 6 107--122.
Mathematical Reviews (MathSciNet): MR2011337
Chiu, S. N. (1997). A central limit theorem for linear Kolmogorov's birth--growth models. Stoch. Proc. Appl. 66 97--106.
Mathematical Reviews (MathSciNet): MR1431872
Digital Object Identifier: doi:10.1016/S0304-4149(96)00113-5
Chiu, S. N. and Quine, M. P. (1997). Central limit theory for the number of seeds in a growth model in $\mathbbR^d$ with inhomogeneous Poisson arrivals. Ann. Appl. Probab. 7 802--814.
Mathematical Reviews (MathSciNet): MR1459271
Digital Object Identifier: doi:10.1214/aoap/1034801254
Project Euclid: euclid.aoap/1034801254
Chiu, S. N. and Quine, M. P. (2001). Central limit theorem for germination-growth models in $\mathbbR^d$ with non-Poisson locations. Adv. in Appl. Probab. 33 751--755.
Mathematical Reviews (MathSciNet): MR1875776
Digital Object Identifier: doi:10.1239/aap/1011994026
Project Euclid: euclid.aap/1011994026
Chiu, S. N. and Lee, H. Y. (2002). A regularity condition and strong limit theorems for linear birth--growth processes. Math. Nachr. 241 21--27.
Mathematical Reviews (MathSciNet): MR1912374
Digital Object Identifier: doi:10.1002/1522-2616(200207)241:1<21::AID-MANA21>3.0.CO;2-D
Devroye, L. (1993). Records, the maximal layer, and uniform distributions in monotone sets. Comput. Math. Appl. 25 19--31.
Mathematical Reviews (MathSciNet): MR1199909
Ehrgott, M. (2000). Multicriteria Optimization. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1788966
Zentralblatt MATH: 0956.90039
Groeneboom, P. (1988). Limit theorems for convex hulls. Probab. Theory Related Fields 79 327--368.
Mathematical Reviews (MathSciNet): MR0959514
Digital Object Identifier: doi:10.1007/BF00342231
Küfer, K. H. (1994). On the approximation of a ball by random polytopes. Adv. in Appl. Probab. 26 876--892.
Mathematical Reviews (MathSciNet): MR1303867
Digital Object Identifier: doi:10.2307/1427895
JSTOR: links.jstor.org
Kolmogorov, A. N. (1937). On statistical theory of metal crystallisation. Izvestia Academy of Science USSR Ser. Math. 3 355--360. (In Russian.)
Møller, J. (1992). Random Johnson--Mehl tessellations. Adv. in Appl. Probab. 24 814--844.
Mathematical Reviews (MathSciNet): MR1188954
Digital Object Identifier: doi:10.2307/1427714
JSTOR: links.jstor.org
Penrose, M. D. (2003). Random Geometric Graphs. Clarendon Press, Oxford.
Mathematical Reviews (MathSciNet): MR1986198
Zentralblatt MATH: 1029.60007
Penrose, M. D. (2007). Gaussian limits for random geometric measures. Electron. J. Probab. To appear.
Mathematical Reviews (MathSciNet): MR2336596
Penrose, M. D. and Yukich, J. E. (2002). Limit theory for random sequential packing and deposition. Ann. Appl. Probab. 12 272--301.
Mathematical Reviews (MathSciNet): MR1890065
Digital Object Identifier: doi:10.1214/aoap/1015961164
Project Euclid: euclid.aoap/1015961164
Penrose, M. D. and Yukich, J. E. (2004). Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 277--303.
Mathematical Reviews (MathSciNet): MR1952000
Digital Object Identifier: doi:10.1214/aoap/1042765669
Project Euclid: euclid.aoap/1042765669
Penrose, M. D. and Yukich, J. E. (2005). Normal approximation in geometric probability. In Stein's Method and Applications. Lecture Notes Ser. Inst. Math. Sci. Nat. Univ. Singap. 5 (A. D. Barbour and L. H. Y. Chen, eds.) 37--58. Singapore Univ. Press. Available at http://www.lehigh.edu/~jey0/publications.html.
Mathematical Reviews (MathSciNet): MR2201885
Pomerol, J.-C. and Barba-Romero, S. (2000). Multicriterion Decision in Management. Kluwer Academic Publishers, Boston.
Zentralblatt MATH: 0986.90014
Reitzner, M. (2002). Random points on the boundary of smooth convex bodies. Trans. Amer. Math. Soc. 354 2243--2278.
Mathematical Reviews (MathSciNet): MR1885651
Digital Object Identifier: doi:10.1090/S0002-9947-02-02962-8
JSTOR: links.jstor.org
Reitzner, M. (2003). Random polytopes and the Efron--Stein jackknife inequality. Ann. Probab. 32 2136--2166.
Mathematical Reviews (MathSciNet): MR2016615
Digital Object Identifier: doi:10.1214/aop/1068646381
Project Euclid: euclid.aop/1068646381
Reitzner, M. (2005). The combinatorial structure of random polytopes. Adv. Math. 191 178--208.
Mathematical Reviews (MathSciNet): MR2102847
Digital Object Identifier: doi:10.1016/j.aim.2004.03.006
Reitzner, M. (2005). Central limit theorems for random polytopes. Probab. Theory Related Fields 133 483--507.
Mathematical Reviews (MathSciNet): MR2197111
Digital Object Identifier: doi:10.1007/s00440-005-0441-8
Rényi, A. (1962). Théorie des éléments saillants d'une suite d'observations. Colloq. on Combinatorial Methods in Prob. Theory 104--115. Math. Institut, Aarhus Universitet, Denmark.
Rényi, A. and Sulanke, R. (1963). Über die konvexe Hülle von $n$ zufállig gewählten Punkten II. Z. Wahrsch. Verw. Gebiete 2 75--84.
Mathematical Reviews (MathSciNet): MR0156262
Digital Object Identifier: doi:10.1007/BF00535300
Resnick, S. I. (1987). Extreme Values, Regular Variation and Point Processes. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR0900810
Zentralblatt MATH: 0633.60001
Schneider, R. (1993). Convex bodies: The Brunn--Minkowski theory. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet): MR1216521
Zentralblatt MATH: 0798.52001
Seneta, E. (1976). Regularly Varying Functions. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR0453936
Zentralblatt MATH: 0324.26002
Vu, V. (2005). Central limit theorems for random polytopes in a smooth convex set. Adv. in Math. 207 221--243.
Mathematical Reviews (MathSciNet): MR2264072
Digital Object Identifier: doi:10.1016/j.aim.2005.11.011
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