Arbitrarily finely divisible stochastic matrices

Abstract

We introduce and study the class of arbitrarily finely divisible stochastic matrices (${\mathrm{AFD}}_{+}$-matrices): stochastic matrices that have a stochastic c-th root for infinitely many natural numbers c. This notion generalises the class of embeddable stochastic matrices. In particular, if A is a transition matrix for a Markov process over some time period, then arbitrary finely divisibility of A inside the set of stochastic matrices is the necessary and sufficient condition for the existence of a transition matrix corresponding to this Markov process over infinitesimally short periods.

Our research explores the connection between the spectral properties of an ${\mathrm{AFD}}_{+}$-matrix A and the spectral properties of a limit point L of its stochastic roots. This connection, which is first formalised in the broader context of complex and real square matrices, poses restrictions on A assuming L is given. For example, if an ${\mathrm{AFD}}_{+}$-matrix A has a corresponding irreducible limit point L, then A has to be a circulant matrix. We identify all matrices that can be a limit point of stochastic roots for some non-singular ${\mathrm{AFD}}_{+}$-matrix. Further, we demonstrate a construction of a class of ${\mathrm{AFD}}_{+}$-matrices with a given limit point L from embeddable matrices. To illustrate these theoretical findings, we examine specific cases, including $2\times 2$ matrices, $3\times 3$ circulant matrices, and offer a complete characterisation of ${\mathrm{AFD}}_{+}$-matrices of rank-two.