Let $\{S_i\}, i = 1, \cdots, n$, be a partition of the circle into sets $S_i$ each consisting of a finite union of arcs. Let $f$ be a rotation of the circle and let $u$ denote Lebesgue measure. Then the matrix $P$ defined by $p_{ij} = u(S_i \cap f^{-1} S_j)/u(S_i)$ is stochastic. We prove (and improve) a conjecture of Joel E. Cohen asserting that every irreducible stochastic matrix arises from a construction of this type.
Publié le : 1983-08-14
Classification:
Measure-preserving transformations,
ergodic theory,
mapping of the unit interval,
Markov chain,
15A51,
28A65,
60J10
@article{1176993523,
author = {Alpern, Steve},
title = {Rotational Representations of Stochastic Matrices},
journal = {Ann. Probab.},
volume = {11},
number = {4},
year = {1983},
pages = { 789-794},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993523}
}
Alpern, Steve. Rotational Representations of Stochastic Matrices. Ann. Probab., Tome 11 (1983) no. 4, pp. 789-794. http://gdmltest.u-ga.fr/item/1176993523/