Rotational Representations of Transition Matrix Functions

Abstract

Let $P(h) = (p_{ij}(h), i, j = 1,2,\ldots, n), h \geq 0, n \geq 1$, be a transition matrix function defining an irreducible recurrent continuous parameter Markov process. Let $(S_i, i = 1,2,\ldots, n)$ be a partition of the circle into sets $S_i$ each consisting of a finite union of arcs $A_{k\ell}$. Let $f_t$ be a rotation of length $t$ of the circle, and denote Lebesgue measure by $\leftthreetimes$. We generalize and prove for the transition matrix function $P(h)$ a theorem of Cohen $(n = 2)$ and Alpern $(n \geq 2)$ asserting that every recurrent stochastic $n \times n$ matrix $P$ is given by \begin{equation*}\tag{*}p_{ij} = \big(\leftthreetimes(S_i\cap f^{-1}_t(S_j)\big)/\leftthreetimes(S_i),\end{equation*} for some choice of rotation $f_t$ and partition $\{S_i\}$. We prove the existence of a continuous map $\Phi$ from the space of $n \times n$ irreducible stochastic matrices into $n$-partitions of [0, 1), such that every domain matrix $P$ is represented by $(\ast)$ with $\{S_i\} = \Phi(P)$ and $t = 1/n$!. Furthermore, the representing process $(f_t, \{S_i\})$ has not only the same transition probabilities but also the same probabilistic cycle distribution as the Markov process based on $P$.