On monotone Markov chains and properties of monotone matrix roots

Monotone matrices are stochastic matrices that satisfy the monotonicity conditions as introduced by Daley in 1968. Monotone Markov chains are useful in modeling phenomena in several areas. Most previous work examines the embedding problem for Markov chains within the entire set of stochastic transition matrices, and only a few studies focus on the embeddability within a specific subset of stochastic matrices. This article examines the embedding in a discrete-time monotone Markov chain, i.e., the existence of monotone matrix roots. Monotone matrix roots of (2 x 2) monotone matrices are investigated in previous work. For (3 x 3) monotone matrices, this article proves properties that are useful in studying the existence of monotone roots. Furthermore, we demonstrate that all (3 x 3) monotone matrices with positive eigenvalues have an mth root that satisfies the monotonicity conditions (for all values m is an element of N, m >= 2). For monotone matrices of order n > 3, diverse scenarios regarding the matrix roots are pointed out, and interesting properties are discussed for block diagonal and diagonalizable monotone matrices.