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Asymptotic methods for random tessellations. (English) Zbl 1296.60019

Spodarev, Evgeny (ed.), Stochastic geometry, spatial statistics and random fields. Asymptotic methods. Selected papers based on the presentations at the summer academy on stochastic geometry, spatial statistics and random fields, Söllerhaus, Germany, September 13–26, 2009. Berlin: Springer (ISBN 978-3-642-33304-0/pbk; 978-3-642-33305-7/ebook). Lecture Notes in Mathematics 2068, 183-204 (2013).
Summary: In this chapter, we are interested in two classical examples of random tessellations which are the Poisson hyperplane tessellation and Poisson-Voronoi tessellation. The first section introduces the main definitions, the application of an ergodic theorem and the construction of the so-called typical cell as the natural object for a statistical study of the tessellation. We investigate a few asymptotic properties of the typical cell by estimating the distribution tails of some of its geometric characteristics (inradius, volume, fundamental frequency). In the second section, we focus on the particular situation where the inradius of the typical cell is large. We start with precise distributional properties of the circumscribed radius that we use afterwards to provide quantitative information about the closeness of the cell to a ball. We conclude with limit theorems for the number of hyperfaces when the inradius goes to infinity.
For the entire collection see [Zbl 1258.60007].

MSC:

60D05 Geometric probability and stochastic geometry
60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
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[1] Avram, F.; Bertsimas, D., On central limit theorems in geometrical probability, Ann. Appl. Probab., 3, 1033-1046 (1993) · Zbl 0784.60015 · doi:10.1214/aoap/1177005271
[2] Baccelli, F.; Blaszczyszyn, B., Stochastic Geometry and Wireless Networks (2009), Delft: Now Publishers, Delft · Zbl 1184.68015
[3] Bandle, C.: Isoperimetric inequalities and applications. Monographs and Studies in Mathematics, vol. 7. Pitman (Advanced Publishing Program), Boston (1980) · Zbl 0436.35063
[4] Bárány, I.; Reitzner, M., On the variance of random polytopes, Adv. Math., 225, 1986-2001 (2010) · Zbl 1204.52007 · doi:10.1016/j.aim.2010.04.012
[5] Bárány, I.; Reitzner, M., Poisson polytopes. Ann. Probab., 38, 1507-1531 (2010) · Zbl 1204.60018 · doi:10.1214/09-AOP514
[6] Bowman, F., Introduction to Bessel functions (1958), New York: Dover, New York · Zbl 0288.60030
[7] Bürgisser, P.; Cucker, F.; Lotz, M., Coverage processes on spheres and condition numbers for linear programming, Ann. Probab., 38, 570-604 (2010) · Zbl 1205.60027 · doi:10.1214/09-AOP489
[8] Calka, P., Mosaïques Poissoniennes de l’espace Euclidian, Une extension d’un résultat de R. E. Miles. C. R. Math. Acad. Sci. Paris, 332, 557-562 (2001) · Zbl 0988.60004
[9] Calka, P., The distributions of the smallest disks containing the Poisson-Voronoi typical cell and the Crofton cell in the plane, Adv. Appl. Probab., 34, 702-717 (2002) · Zbl 1029.60002 · doi:10.1239/aap/1037990949
[10] Calka, P.; Kendall, WS; Molchanov, I., Tessellations, New Perspectives in Stochastic Geometry (2010), London: Oxford Univerxity Press, London · Zbl 1183.60001
[11] Calka, P.; Schreiber, T., Limit theorems for the typical Poisson-Voronoi cell and the Crofton cell with a large inradius, Ann. Probab., 33, 1625-1642 (2005) · Zbl 1084.60008 · doi:10.1214/009117905000000134
[12] Calka, P.; Schreiber, T., Large deviation probabilities for the number of vertices of random polytopes in the ball, Adv. Appl. Probab., 38, 47-58 (2006) · Zbl 1099.60009 · doi:10.1239/aap/1143936139
[13] Calka, P.; Schreiber, T.; Yukich, JE, Brownian limits, local limits and variance asymptotics for convex hulls in the ball (2012), Probab: Ann, Probab
[14] Cowan, R.: The use of the ergodic theorems in random geometry. Adv. in Appl. Prob. 10 (suppl.), 47-57 (1978) · Zbl 0381.60012
[15] Cowan, R., Properties of ergodic random mosaic processes, Math. Nachr., 97, 89-102 (1980) · Zbl 0475.60010 · doi:10.1002/mana.19800970109
[16] Dryden, IL; Farnoosh, R.; Taylor, CC, Image segmentation using Voronoi polygons and MCMC, with application to muscle fibre images, J. Appl. Stat., 33, 609-622 (2006) · Zbl 1118.62342 · doi:10.1080/02664760600679825
[17] Feller, W., An Introduction to Probability Theory and its Applications (1971), New York: Wiley, New York · Zbl 0219.60003
[18] Fleischer, F.; Gloaguen, C.; Schmidt, H.; Schmidt, V.; Schweiggert, F., Simulation algorithm of typical modulated Poisson-Voronoi cells and application to telecommunication network modelling, Jpn. J. Indust. Appl. Math., 25, 305-330 (2008) · Zbl 1166.90005 · doi:10.1007/BF03168553
[19] Foss, S.; Zuyev, S., On a Voronoi aggregative process related to a bivariate Poisson process, Adv. Appl. Probab., 28, 965-981 (1996) · Zbl 0867.60004 · doi:10.2307/1428159
[20] Gerstein, M.; Tsai, J.; Levitt, M., The volume of atoms on the protein surface: calculated from simulation, using Voronoi polyhedra, J. Mol. Biol., 249, 955-966 (1995) · doi:10.1006/jmbi.1995.0351
[21] Gilbert, EN, Random subdivisions of space into crystals, Ann. Math. Stat., 33, 958-972 (1962) · Zbl 0242.60009 · doi:10.1214/aoms/1177704464
[22] Gilbert, EN, The probability of covering a sphere with n circular caps, Biometrika, 52, 323-330 (1965) · Zbl 0137.36202
[23] Goldman, A., Le spectre de certaines mosaïques Poissoniennes du plan et l’enveloppe convexe du pont Brownien, Probab. Theor. Relat. Fields, 105, 57-83 (1996) · Zbl 0858.35094 · doi:10.1007/BF01192071
[24] Goldman, A., Sur une conjecture de D, G. Kendall concernant la cellule de Crofton du plan et sur sa contrepartie brownienne. Ann. Probab., 26, 1727-1750 (1998) · Zbl 0936.60009
[25] Goldman, A.; Calka, P., On the spectral function of the Poisson-Voronoi cells, Ann. Inst. H. Poincaré Probab. Stat., 39, 1057-1082 (2003) · Zbl 1031.60009 · doi:10.1016/S0246-0203(03)00025-6
[26] Goudsmit, S., Random distribution of lines in a plane, Rev. Mod. Phys., 17, 321-322 (1945) · Zbl 0063.01715 · doi:10.1103/RevModPhys.17.321
[27] Hall, P., On the coverage of k-dimensional space by k-dimensional spheres, Ann. Probab., 13, 991-1002 (1985) · Zbl 0582.60015 · doi:10.1214/aop/1176992920
[28] Heinrich, L.; Schmidt, H.; Schmidt, V., Central limit theorems for Poisson hyperplane tessellations, Ann. Appl. Probab., 16, 919-950 (2006) · Zbl 1132.60023 · doi:10.1214/105051606000000033
[29] Hug, D.; Baddeley, AJ; Bárány, I.; Schneider, R.; Weil, W., Random mosaics, Stochastic Geometry - Lecture Notes in Mathematics (2007), Berlin: Springer, Berlin
[30] Hug, D.; Reitzner, M.; Schneider, R., Large Poisson-Voronoi cells and Crofton cells, Adv. Appl. Probab., 36, 667-690 (2004) · Zbl 1029.60008 · doi:10.1239/aap/1093962228
[31] Hug, D.; Reitzner, M.; Schneider, R., The limit shape of the zero cell in a stationary Poisson hyperplane tessellation, Ann. Probab., 32, 1140-1167 (2004) · Zbl 1050.60010 · doi:10.1214/aop/1079021474
[32] Hug, D.; Schneider, R., Asymptotic shapes of large cells in random tessellations, Geom. Funct. Anal., 17, 156-191 (2007) · Zbl 1114.60012 · doi:10.1007/s00039-007-0592-0
[33] Kac, M., Can one hear the shape of a drum?, Am. Math. Mon., 73, 1-23 (1966) · Zbl 0139.05603 · doi:10.2307/2313748
[34] Kovalenko, IN, A simplified proof of a conjecture of D, G. Kendall concerning shapes of random polygons. J. Appl. Math. Stoch. Anal., 12, 301-310 (1999) · Zbl 0959.60007 · doi:10.1155/S1048953399000283
[35] Last, G., Stationary partitions and Palm probabilities, Adv. Appl. Probab., 38, 602-620 (2006) · Zbl 1121.60008 · doi:10.1239/aap/1158684994
[36] Lautensack, C.; Zuyev, S., Random Laguerre tessellations. Adv. Appl. Probab., 40, 630-650 (2008) · Zbl 1154.60011 · doi:10.1239/aap/1222868179
[37] Maier, R.; Schmidt, V., Stationary iterated tessellations, Adv. Appl. Probab., 35, 337-353 (2003) · Zbl 1041.60012 · doi:10.1239/aap/1051201649
[38] Mecke, J., Stationäre zufällige Maße auf lokalkompakten Abelschen Gruppen, Z. Wahrscheinlichkeitstheorie und verw. Gebiete, 9, 36-58 (1967) · Zbl 0164.46601 · doi:10.1007/BF00535466
[39] Mecke, J., On the relationship between the 0-cell and the typical cell of a stationary random tessellation, Pattern Recogn., 32, 1645-1648 (1999) · doi:10.1016/S0031-3203(99)00026-6
[40] Miles, RE, Random polygons determined by random lines in a plane, Proc. Natl. Acad. Sci. USA, 52, 901-907 (1964) · Zbl 0122.17301 · doi:10.1073/pnas.52.4.901
[41] Miles, RE, Random polygons determined by random lines in a plane II, Proc. Natl. Acad. Sci. USA, 52, 1157-1160 (1964) · Zbl 0133.16001 · doi:10.1073/pnas.52.5.1157
[42] Miles, RE, The various aggregates of random polygons determined by random lines in a plane, Adv. Math., 10, 256-290 (1973) · Zbl 0292.60020 · doi:10.1016/0001-8708(73)90110-2
[43] Møller, J., Random Johnson-Mehl tessellations. Adv. Appl. Probab., 24, 814-844 (1992) · Zbl 0768.60014 · doi:10.2307/1427714
[44] Møller, J.: Lectures on random Voronoi tessellations. In: Lecture Notes in Statistics, vol. 87. Springer, New York (1994) · Zbl 0812.60016
[45] Müller, A.; Stoyan, D., Comparison Methods for Stochastic Models and Risks (2002), Chichester: Wiley, Chichester · Zbl 0999.60002
[46] Nagaev, AV, Some properties of convex hulls generated by homogeneous Poisson point processes in an unbounded convex domain, Ann. Inst. Stat. Math., 47, 21-29 (1995) · Zbl 0829.60040 · doi:10.1007/BF00773409
[47] Nagel, W.; Weiss, V., Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration, Adv. Appl. Probab., 37, 859-883 (2005) · Zbl 1098.60012 · doi:10.1239/aap/1134587744
[48] Okabe, A.; Boots, B.; Sugihara, K.; Chiu, SN, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams (2000), Wiley, Chichester: With a foreword by D. G. Kendall, Wiley, Chichester · Zbl 0946.68144 · doi:10.1002/9780470317013
[49] Paroux, K., Quelques théorèmes centraux limites pour les processus Poissoniens de droites dans le plan, Adv. Appl. Probab., 30, 640-656 (1998) · Zbl 0914.60013 · doi:10.1239/aap/1035228121
[50] Reitzner, M., Random polytopes and the Efron-Stein jackknife inequality, Ann. Probab., 31, 2136-2166 (2003) · Zbl 1058.60010 · doi:10.1214/aop/1068646381
[51] Reitzner, M., Central limit theorems for random polytopes, Probab. Theor. Relat. Fields, 133, 483-507 (2005) · Zbl 1081.60008 · doi:10.1007/s00440-005-0441-8
[52] Rényi, A.; Sulanke, R., Über die konvexe Hülle von n zufällig gewählten Punkten, Z. Wahrscheinlichkeitstheorie und verw. Gebiete, 2, 75-84 (1963) · Zbl 0118.13701 · doi:10.1007/BF00535300
[53] Schneider, R.; Weil, W., Stochastic and Integral Geometry (2008), Berlin: Springer, Berlin · Zbl 1175.60003 · doi:10.1007/978-3-540-78859-1
[54] Schreiber, T., Variance asymptotics and central limit theorems for volumes of unions of random closed sets, Adv. Appl. Probab., 34, 520-539 (2002) · Zbl 1018.60012 · doi:10.1239/aap/1033662164
[55] Schreiber, T., Asymptotic geometry of high density smooth-grained Boolean models in bounded domains, Adv. Appl. Probab., 35, 913-936 (2003) · Zbl 1041.60014 · doi:10.1239/aap/1067436327
[56] Schreiber, T.; Yukich, JE, Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points, Ann. Probab., 36, 363-396 (2008) · Zbl 1130.60031 · doi:10.1214/009117907000000259
[57] Shepp, L., Covering the circle with random arcs, Israel J. Math., 11, 328-345 (1972) · Zbl 0241.60008 · doi:10.1007/BF02789327
[58] Siegel, AF; Holst, L., Covering the circle with random arcs of random sizes, J. Appl. Probab., 19, 373-381 (1982) · Zbl 0489.60017 · doi:10.2307/3213488
[59] Stevens, WL, Solution to a geometrical problem in probability, Ann. Eugenics, 9, 315-320 (1939) · Zbl 0023.05603 · doi:10.1111/j.1469-1809.1939.tb02216.x
[60] Stoyan, D.; Kendall, W.; Mecke, J., Stochastic Geometry and Its Applications (1995), New York: Wiley, New York · Zbl 0838.60002
[61] Vu, V., Sharp concentration of random polytopes, Geom. Funct. Anal., 15, 1284-1318 (2005) · Zbl 1094.52002 · doi:10.1007/s00039-005-0541-8
[62] Weygaert, R., Fragmenting the universe III. The construction and statistics of 3-D Voronoi tessellations, Astron. Astrophys., 283, 361-406 (1994)
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