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The purpose of the paper is to establish a large deviation principle for a certain class of increasing set-valued processes obeyingMarkovian dynamics. The obtained result is then applied to investigate the asymptotics of the sequence of successive convex hulls generated by uniform samples on a d-dimensional ball.
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Tom
Strony
273--285
Opis fizyczny
Bibliogr. 18 poz.
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Bibliografia
- [1] H. Braker and T. Hsing, On the area and perimeter of a random convex hull in a bounded convex set, Probab. Theory Related Fields 111 (1998), pp. 517-550.
- [2] A. J. Cabo and P. Groeneboom, Limit theorems for functionals of convex hulls, Probab. Theory Related Fields 100 (1994), pp. 31-55.
- [3] J.-D. Deuschel and D. W. Stroock, Large Deviations, Academic Press, Inc., Boston 1989.
- [4] P. Dupuis and R. S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, Wiley, Chichester 1997.
- [5] B. Efron, The convex hull of a random set of points, Biometrika 52 (1965), pp. 331-343.
- [6] P. Groeneboom, Limit theorems for convex hulls, Probab. Theory Related Fields 79 (1988), pp. 327-368.
- [7] T. Hsing, On the asymptotic distribution of the area outside a random convex hull in a disk, Ann. Appl. Probab. 4 (2) (1994), pp. 478-493.
- [8] W. S. Kendall, J. Meeke and D. Stoyan, Stochastic Geometry and Its Applications, Akademie-Verlag,. Berlin 1987.
- [9] I. M. Khamdamov and A. V. Nagaev, Limiting distributions for functionals of the convex hull generated by uniformly distributed variables (in Russian), Dokl. Akad. Nauk UzSSR, Tashkent, 7 (1991), pp. 8-9.
- [10] K. H. Küfer, On the approximation of a ball by random polytopes, Adv. Appl. Probab. 26 (1994), pp. 876-892.
- [11] G. Matheron, Random Sets and Integral Geometry, Wiley, New York 1975.
- [12] I. S. Molchanov, Limit Theorems for Unions of Random Closed Sets, Lecture Notes in Math. 1561, Springer, Berlin 1993.
- [13] I. S. Molchanov, On the convergence of random processes generated by polyhedral approximation of ^convex compacts, Theory Probab. Appl. 40 (1995), pp. 383-390.
- [14] G. L. O’Brien and W. Yerwaat, Capacities, large deviations and loglog laws, in: Stable Processes and Related Topics, Progress in Probability, Vol. 25, Birkhäuser, Boston 1991, pp. 43-83.
- [15] A. A. Pukhalskii, On functional principle of large deviations, in: New Trends in Probability and Statistics, V. Sazonov and T. Shervashidze (Eds.), VSP-Mokslal, Zeist, The Netherlands, 1991, pp. 198-218.
- [16] A. Rényi and R. Sulanke, Über die konvexe Hülle von n zufällig gewählten Punkten (I and II), Z. Wahrscheinlichkeitstheorie verw. Gebiete 2 (1963), pp. 75-84; ibidem 3 (1964), pp. 138-147.
- [17] R. Schneider, Random approximation of convex sets, Journal of Microscopy 151 (1988), pp. 211-227.
- [18] R. Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopaedia of Mathematics and Its Applications 44, Cambridge University Press, 1993.
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Bibliografia
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