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Invariant distributions for shapes in sequences of randomly-divided rectangles

  • Francis K. C. Chen (a1) and Richard Cowan (a2)

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.

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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.

References

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[2] 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/).
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[9] 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. 110123.
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Invariant distributions for shapes in sequences of randomly-divided rectangles

  • Francis K. C. Chen (a1) and Richard Cowan (a2)

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