A Geometric Representation of a Stochastic Matrix: Theorem and Conjecture

Abstract

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?