An irreducible stochastic matrix may be constructed by partitioning a line of unit length into a finite number of intervals, shifting the line to the right $(\mod 1)$ by a small amount, and defining transition probabilities in terms of the overlaps among the intervals before and after the shift. It is proved that every $2 \times 2$ irreducible stochastic matrix arises from this construction. Does every $n \times n$ irreducible stochastic matrix arise this way?
Publié le : 1981-10-14
Classification:
Measure-preserving transformation,
ergodic theory,
mapping of the unit interval,
Markov chain,
15A51,
28A65,
60J10
@article{1176994319,
author = {Cohen, Joel E.},
title = {A Geometric Representation of a Stochastic Matrix: Theorem and Conjecture},
journal = {Ann. Probab.},
volume = {9},
number = {6},
year = {1981},
pages = { 899-901},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994319}
}
Cohen, Joel E. A Geometric Representation of a Stochastic Matrix: Theorem and Conjecture. Ann. Probab., Tome 9 (1981) no. 6, pp. 899-901. http://gdmltest.u-ga.fr/item/1176994319/