Arbitrarily Finely Divisible Matrices

TL;DR

This paper introduces and studies arbitrarily finely divisible matrices, a class of stochastic matrices that can be rooted infinitely often, which is crucial for understanding Markov processes over arbitrarily short time intervals.

Contribution

It defines the class of arbitrarily finely divisible matrices, extending the concept of infinite divisibility, and provides foundational analysis with specific matrix examples.

Findings

Characterization of arbitrarily finely divisible matrices

Connection to Markov process transition matrices

Examples with 2x2, circulant, and rank-two matrices

Abstract

The class of stochastic matrices that have a stochastic $c$-th root for infinitely many natural numbers $c$ is introduced and studied. Such matrices are called arbitrarily finely divisible, and generalise the class of infinitely divisible matrices. In particular, if $A$ is a transition matrix for a Markov process over some time period, then arbitrarily finely divisibility of $A$ is the necessary and sufficient condition for the existence of transition matrices corresponding to this Markov process over arbitrarily short periods. In this paper, we lay the foundation for research into arbitrarily finely divisible matrices and demonstrate the concepts using specific examples of $2 \times 2$ matrices, $3 \times 3$ circulant matrices, and rank-two matrices.