Skip to main content
Article thumbnail
Location of Repository

Rotational (and other) representations of stochastic matrices

By Steven Alpern and V. S. Prasad

Abstract

Joel E. Cohen (1981) conjectured that any stochastic matrix P = fpi;jg could be represented by some circle rotation f in the following sense: For some par- tition fSig of the circle into sets consisting of nite unions of arcs, we have (*) pi;j = (f (Si) \ Sj) = (Si), where denotes arc length. In this paper we show how cycle decomposition techniques originally used (Alpern, 1983) to establish Cohen�s conjecture can be extended to give a short simple proof of the Coding Theorem, that any mixing (that is, PN > 0 for some N) stochastic matrix P can be represented (in the sense of * but with Si merely measurable) by any aperiodic measure preserving bijection (automorphism) of a Lesbesgue proba- bility space. Representations by pointwise and setwise periodic automorphisms are also established. While this paper is largely expository, all the proofs, and some of the results, are new

Topics: QA Mathematics
Publisher: Centre for Discrete and Applicable Mathematics, London School of Economics and Political Science
Year: 2005
OAI identifier: oai:eprints.lse.ac.uk:13931
Provided by: LSE Research Online
Download PDF:
Sorry, we are unable to provide the full text but you may find it at the following location(s):
  • http://www.cdam.lse.ac.uk (external link)
  • http://eprints.lse.ac.uk/13931... (external link)
  • Suggested articles


    To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.