Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-01T09:08:38.542Z Has data issue: false hasContentIssue false
  • English
  • Français

Shapes of Rectangular Prisms After Repeated Random Division

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

Published online by Cambridge University Press:  01 July 2016

Richard Cowan
Richard Cowan*
Affiliation:
University of Hong Kong
*
∗ Postal address: Department of Statistics, University of Hong Kong, Pokfulam Road, Hong Kong. Email: rcowan@hkursc.hku.hk
Get access Rights & Permissions [Opens in a new window]

Abstract

The shape of a rectangular prism in (d + 1)-dimensions is defined as Y = (Y1, Y2, · ··, Yd), Yn = Ln/Ln+1 where the Ln are the prism's edge lengths, in ascending order. We investigate shape distributions that are invariant when the prism is cut into two, also rectangular, prisms, with one prism retained for measurement and the other discarded. Interesting new distributions on [0, 1]d arise.

Keywords

INVARIANT DISTRIBUTIONSSHAPE DISTRIBUTIONSRANDOM GEOMETRYDISTRIBUTION THEORYMARKOV PROCESSES

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 29 , Issue 1 , March 1997 , pp. 26 - 37
Copyright
Copyright © Applied Probability Trust 1997 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bookstein, F. (1978) The Measurement of Biological Shape and Shape Change (Lecture Notes in Biomathematics 24). Springer, New York.Google Scholar
Kendall, D. G. (1977) The diffusion of shape. Adv. Appl. Prob. 9, 428–430.Google Scholar
Kendall, D. G. (1984) Shape manifolds, procrustean metrics and complex projective spaces. Bull. London Math. Soc. 16, 81–121.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. 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. (1986) Random triangles. In Proc. Conf. Stochastic Geometry. ed. Barndorff-Nielsen, O.. Aarhus.Google Scholar