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Invariant distributions for shapes in sequences of randomly-divided rectangles

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

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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