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On monotone Markov chains and properties of monotone matrix roots

Journal Contribution - Journal Article

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 focuses on the embeddability within a specific subset of stochastic matrices. The present paper examines for discrete-time monotone Markov chains 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, the present paper 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 m-th root that satisfies the monotonicity conditions (for all natural numbers m > 1). 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 for diagonalizable monotone matrices.
Journal: Special Matrices
ISSN: 2300-7451
Issue: 1
Volume: 11
Pages: 1-14
Publication year:2022
Keywords:monotone Markov chain, monotone matrix, matrix roots, embedding problem
Accessibility:Open