Advances in Applied Probability

Invariant distributions for shapes in sequences of randomly-divided rectangles

Francis K. C. Chen and Richard Cowan

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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.

Article information

Adv. in Appl. Probab. Volume 31, Number 1 (1999), 1-14.

First available in Project Euclid: 21 August 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60E05: Distributions: general theory 60J05: Discrete-time Markov processes on general state spaces

Invariant distributions shape stochastic geometry distribution theory Markov processes dynamical systems


Chen, Francis K. C.; Cowan, Richard. Invariant distributions for shapes in sequences of randomly-divided rectangles. Adv. in Appl. Probab. 31 (1999), no. 1, 1--14. doi:10.1239/aap/1029954262.

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