Shape-driven nested Markov tessellations

Stochastics

Volumn 85, Issue 3, 2013, Pages 510-531

  Schreiber, Tomasz ^a   Thäle, Christoph ^b  

^a NICOLAUS COPERNICUS UNIVERSITY   (Poland)

^b UNIVERSITY OF OSNABRÜCK   (Germany)

Author keywords

iteration; Markov process; mean values; nested tessellation; random tessellation; stochastic geometry

Indexed keywords

EID: 84879061224     PISSN: 17442508     EISSN: 17442516     Source Type: Journal    
DOI: 10.1080/17442508.2011.654344     Document Type: Article

References (25)

1. Bertoin, J. 2006. Random Fragmentation and Coagulation Processes, Cambridge: Cambridge University Press.
2. Cowan, R. 2010. New classes of random tessellations arising from iterative divisions of cells. Adv. Appl. Probab., 42: 26 - 47.
3. Daley, D.J. and Vere-Jones, D. 1988. An Introduction to the Theory of Point Processes, New York: Springer.
4. Dereudre, D. 2008. Gibbs Delaunay tessellations with geometric hardcore condition. J. Statist. Phys., 121: 511 - 551.
5. Dereudre, D., Drouilhet, R. and Georgii, H.-O. Existence of Gibbs point processes with geometry-dependent interactions. Probab. Theor. Relat. Fields, to appear
6. Ethier, S.N. and Kurtz, T.G. 1986. Markov Processes - Characterization and Convergence, Chichester: Wiley.
7. Gruber, P.M. 2007. Convex and Discrete Geometry, New York: Springer.
8. Kallenberg, O. 2002. Foundations of Modern Probability, 2nd ed., New York: Springer.
9. Liggett, T.M. 2010. Continuous Time Markov Processes: An Introduction, Vol. 113, AMS Graduate, Studies Mathematics.
10. Mecke, J., Nagel, W. and Weiss, V. 2008. A global construction of homogeneous random planar tessellations that are stable under iteration. Stochastics, 80: 51 - 67.
11. Mecke, J., Nagel, W. and Weiss, V. 2008. The iteration of random tessellations and a construction of a homogeneous process of cell division. Adv. Appl. Probab., 40: 49 - 59.
12. Molchanov, I.S. 2005. Theory of Random Sets, Berlin: Springer.
13. Nagel, W. and Weiss, V. 2005. Crack STIT tessellations: Characterization of stationary random tessellations stable with respect to iteration. Adv. Appl. Probab., 37: 859 - 883.
14. Nagel, W. and Weiss, V. 2006. STIT tessellations in the plane. Rendiconti del circulo matematico di Palermo, Serie II (Suppl. 77): 441 - 458.
15. Nagel, W. and Weiss, V. 2008. Mean values for homogeneous STIT tessellations in 3D. Image Anal. Stereol., 27: 29 - 37.
16. Schmidt, H. 2008. Asymptotic Analysis of Stationary Random Tessellations: Applications in Network, Modelling, Saarbrücken: VDM Verlag Dr. Müller.
17. Schneider, R. and Weil, W. 2008. Stochastic and Integral Geometry, Berlin: Springer.
18. Schreiber, T., Georgii, H.-O. and Thäle, C. Entropy and large deviations for iteration stable and general nested tessellations. in preparation
19. Schreiber, T. and Thäle, C. 2010. Typical geometry, second-order properties and central limit theory for iteration stable tessellations. arXiv: 1001.0990 [math.PR]
20. Schreiber, T. and Thäle, C. 2010. Second-order properties and central limit theory for the vertex process of iteration infinitely divisible and iteration stable random tessellations in the plane. Adv. Appl. Probab., 42: 913 - 935.
21. Schreiber, T. and Thäle, C. 2011. Intrinsic volumes of the maximal facet process in higher dimensional STIT tessellations. Stoch. Proc. Appl., 121: 989 - 1012.
22. Stoyan, D., Kendall, D.G. and Mecke, J. 1995. Stochastic Geometry, 2nd ed., Chichester: Wiley.
23. Thäle, C. and Weiss, V. 2010. New mean values for spatial homogeneous random tessellations stable under iteration. Image Anal. Stereol., 29: 143 - 157.
24. Weiss, V. and Cowan, R. 2011. Topological relationships in spatial tessellations. Adv. Appl. Probab., 43: 963 - 984.
25. Weiss, V., Ohser, J. and Nagel, W. 2010. Second moment measure and K-function for planar STIT tessellations. Image Anal. Stereol., 29: 121 - 131.