Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-24T06:59:26.166Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence to collinearity of a sequence of random triangle shapes

Published online by Cambridge University Press:  01 July 2016

David Mannion
David Mannion*
Affiliation:
Royal Holloway and Bedford New College
*
∗Postal address: Department of Mathematics, Royal Holloway and Bedford New College, Egham Hill, Egham, Surrey, TW200EX, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

A sequence of random triangles is constructed by choosing successively the three vertices of one triangle at random in the interior of its predecessor. A way is found to prove that the shapes of these triangles converge, almost surely, to collinear shapes, thus closing a gap in one of the central arguments of Mannion [5]. The new approach is based on a representation of the triangle process by a sequence of products of i.i.d. random matrices. We succeed in calculating the corresponding Lyapounov exponent.

Keywords

SHAPE-SPACESIMPLEXCONTENT OF SIMPLEXPRODUCTS OF RANDOM MATRICESLYAPOUNOV EXPONENT
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 4 , December 1990 , pp. 831 - 844
Copyright
Copyright © Applied Probability Trust 1990 

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

1. Billingsley, P. (1979) Probability and Measure. Wiley, New York.Google Scholar
2. Bougerol, P. and Lacroix, J. (1985) Products of Random Matrices with Applications to Schrödinger Operators. Birkhauser, Boston.CrossRefGoogle Scholar
3. Furstenberg, H. (1963) Non-commuting random products. Trans. Amer. Math. Soc. 108, 377–428.CrossRefGoogle Scholar
4. Kendall, D. G. (1985) Exact distributions for shapes of random triangles in convex sets. Adv. Appl. Prob. 17, 308–329.Google Scholar
5. Mannion, D. (1988) A Markov chain of triangle shapes. Adv. Appl. Prob. 20, 348–370.CrossRefGoogle Scholar
6. Mannion, D. (1990) The invariant distribution of a sequence of random collinear triangle shapes. Adv. Appl. Prob. 22, 845–865.Google Scholar
7. Pollard, D. (1984) Convergence of Stochastic Processes. Springer-Verlag, New York.Google Scholar