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Keywords:
hypergeometric function; iteration/nesting; random tessellation; segments; stochastic geometry; stochastic stability
hypergeometric function; iteration/nesting; random tessellation; segments; stochastic geometry; stochastic stability
Summary:
A result about the distribution of the number of nodes in the relative interior of the typical I-segment in homogeneous and isotropic random tessellations stable under iteration (STIT tessellations) is extended to the anisotropic case using recent findings from Schreiber/Thäle, Typical geometry, second-order properties and central limit theory for iteration stable tessellations, arXiv:1001.0990 [math.PR] (2010). Moreover a new expression for the values of this probability distribution is presented in terms of the Gauss hypergeometric function ${_2F_1}$.
References:
[1] Abramowitz M., Stegun I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1965, online version under http://www.math.ucla.edu/ cbm/aands/index.htm. MR 1225604 | Zbl 0643.33001
[2] Mecke J., Nagel W., Weiss V.: Length distributions of edges in planar stationary and isotropic STIT tessellations. J. Contemp. Math. Anal. 42 (2007), 28–43. DOI 10.3103/S1068362307010025 | MR 2361580 | Zbl 1155.60005
[3] Mecke J., Nagel W., Weiss V.: Some distributions for I-segments of planar random homogeneous STIT tessellations. Math. Nachr. (2010)(to appear). MR 2832660
[4] Nagel W., Weiss V.: Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration. Adv. in Appl. Probab. 37 (2005), 859–883. DOI 10.1239/aap/1134587744 | MR 2193987 | Zbl 1098.60012
[5] Schneider R., Weil W.: Stochastic and Integral Geometry. Springer, Berlin, 2008. MR 2455326 | Zbl 1175.60003
[6] Schreiber T., Thäle C.: Typical geometry, second-order properties and central limit theory for iteration stable tessellations. arXiv:1001.0990 [math.PR] (2010). MR 2796670
[7] Thäle C.: Moments of the length of line segments in homogeneous planar STIT tessellations. Image Anal. Stereol. 28 (2009), 69–76. DOI 10.5566/ias.v28.p69-76 | MR 2538063
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