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Invariant distributions for shapes in sequences of randomly-divided rectangles

Part of: Markov processes Geometric probability and stochastic geometry Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Francis K. C. Chen  and
Richard Cowan
Francis K. C. Chen*
Affiliation:
University of Hong Kong
Richard Cowan*
Affiliation:
University of Sydney
*
∗ Postal address: Department of Statistics, University of Hong Kong, Pokfulam Road, Hong Kong.
∗∗ Postal address: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia.
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Abstract

Interest has been shown in Markovian sequences of geometric shapes. Mostly the equations for invariant probability measures over shape space are extremely complicated and multidimensional. This paper deals with rectangles which have a simple one-dimensional shape descriptor. We explore the invariant distributions of shape under a variety of randomised rules for splitting the rectangle into two sub-rectangles, with numerous methods for selecting the next shape in sequence. Many explicit results emerge. These help to fill a vacant niche in shape theory, whilst contributing at the same time, new distributions on [0,1] and interesting examples of Markov processes or, in the language of another discipline, of stochastic dynamical systems.

Keywords

Invariant distributionsshapestochastic geometrydistribution theoryMarkov processesdynamical systems

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60E05: Distributions 60J05: Discrete-time Markov processes on general state spaces
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 1 , March 1999 , pp. 1 - 14
Copyright
Copyright © Applied Probability Trust 1999 

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References

Arnold, L. and Crauel, H. (1991). Random dynamical systems. In Lyapunov Exponents, eds. Arnold, L., Crauel, H. and Eckmann, J.-P. (Lecture Notes in Math. 1486). Springer, Berlin.Google Scholar
Chen, F. K. C. and Cowan, R. (1997). Proofs of Invariant Distributions for Shapes in Sequences of Randomly-divided Rectangles. Research Report 166, Department of Statistics, University of Hong Kong (available at http://www.maths.usyd.edu.au:8000/u/richardc/).Google Scholar
Cowan, R. (1997). Shapes of rectangular prisms after repeated random division. Adv. Appl. Prob 29, 26–37.Google Scholar
Kendall, D. G. (1977). The diffusion of shape. Adv. Appl. Prob. 9, 428–430.CrossRefGoogle Scholar
Kendall, D. G. (1984). Shape manifolds, procrustean metrics and complex projective spaces. Bull. London Math. Soc. 16, 81–121.Google Scholar
Le, H. and Cowan, R. (1997). A proof, based on stochastic ordering, for degenerate convergence of shapes. Research Report 148, Department of Statistics, University of Hong Kong.Google Scholar
Mannion, D. (1990). Convergence to collinearity of a sequence of random triangle shapes. Adv. Appl. Prob. 22, 831–844.CrossRefGoogle Scholar
Mannion, D. (1993). Products of 2×2 random matrices. Ann. Appl. Prob. 3, 1189–1218.Google Scholar
Miles, R. E. (1983). On the repeated splitting of a planar domain. In Proc. Oberwolfach Conference on Stochastic Geometry, Geometric Statistics and Stereology, ed. Ambartzumian, R. and Weil, W. Teubner, Leipzig, pp. 110–123.Google Scholar
Orey, S. (1971). Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand Reinhold, London.Google Scholar
Watson, G. S. (1986). The shapes of a random sequence of triangles. Adv. Appl. Prob. 18, 156–169.Google Scholar