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On monotonically proceeding structures and stepwise increasing transition matrices of Markov chains

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Abstract
In general, the transition matrix of a Markov chain is a stochastic matrix. For a system that is modeled by a Markov chain, the transition matrix must reflect the characteristics of that system. The present paper introduces a particular class of transition matrices in order to model Markov systems for which, as the length of the time interval becomes greater, a transition from one state to another is more likely. We call these transition matrices stepwise increasing. Moreover in some contexts it is less desirable that the stocks fluctuate over time. In those situations, one is interested in monotonically proceeding stock vectors. This paper examines monotonically proceeding stock vectors and stepwise increasing transition matrices. We present conditions on the transition matrix such that all stock vectors are monotonically proceeding. In particular, the set of monotonically proceeding vectors is characterized for the two-state and three-state cases.
Keywords
Statistics and Probability. Markov chain, transition matrix, eigenvalue, eigenvector, monotonic evolution

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MLA
Guerry, Marie-Anne, and Philippe Carette. “On Monotonically Proceeding Structures and Stepwise Increasing Transition Matrices of Markov Chains.” COMMUNICATIONS IN STATISTICS-THEORY AND METHODS, vol. 51, no. 1, 2022, pp. 51–67, doi:10.1080/03610926.2020.1742921.
APA
Guerry, M.-A., & Carette, P. (2022). On monotonically proceeding structures and stepwise increasing transition matrices of Markov chains. COMMUNICATIONS IN STATISTICS-THEORY AND METHODS, 51(1), 51–67. https://doi.org/10.1080/03610926.2020.1742921
Chicago author-date
Guerry, Marie-Anne, and Philippe Carette. 2022. “On Monotonically Proceeding Structures and Stepwise Increasing Transition Matrices of Markov Chains.” COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 51 (1): 51–67. https://doi.org/10.1080/03610926.2020.1742921.
Chicago author-date (all authors)
Guerry, Marie-Anne, and Philippe Carette. 2022. “On Monotonically Proceeding Structures and Stepwise Increasing Transition Matrices of Markov Chains.” COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 51 (1): 51–67. doi:10.1080/03610926.2020.1742921.
Vancouver
1.
Guerry M-A, Carette P. On monotonically proceeding structures and stepwise increasing transition matrices of Markov chains. COMMUNICATIONS IN STATISTICS-THEORY AND METHODS. 2022;51(1):51–67.
IEEE
[1]
M.-A. Guerry and P. Carette, “On monotonically proceeding structures and stepwise increasing transition matrices of Markov chains,” COMMUNICATIONS IN STATISTICS-THEORY AND METHODS, vol. 51, no. 1, pp. 51–67, 2022.
@article{8655083,
  abstract     = {{In general, the transition matrix of a Markov chain is a stochastic matrix. For a system that is modeled by a Markov chain, the transition matrix must reflect the characteristics of that system. The present paper introduces a particular class of transition matrices in order to model Markov systems for which, as the length of the time interval becomes greater, a transition from one state to another is more likely. We call these transition matrices stepwise increasing. Moreover in some contexts it is less desirable that the stocks fluctuate over time. In those situations, one is interested in monotonically proceeding stock vectors. This paper examines monotonically proceeding stock vectors and stepwise increasing transition matrices. We present conditions on the transition matrix such that all stock vectors are monotonically proceeding. In particular, the set of monotonically proceeding vectors is characterized for the two-state and three-state cases.}},
  author       = {{Guerry, Marie-Anne and Carette, Philippe}},
  issn         = {{0361-0926}},
  journal      = {{COMMUNICATIONS IN STATISTICS-THEORY AND METHODS}},
  keywords     = {{Statistics and Probability. Markov chain,transition matrix,eigenvalue,eigenvector,monotonic evolution}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{51--67}},
  title        = {{On monotonically proceeding structures and stepwise increasing transition matrices of Markov chains}},
  url          = {{http://doi.org/10.1080/03610926.2020.1742921}},
  volume       = {{51}},
  year         = {{2022}},
}

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