@article {
author = {Hirose, Kenichi},
title = {On the spectral properties and convergence of the bonus-malus Markov chain model},
journal = {Journal of Mathematical Modeling},
volume = {9},
number = {4},
pages = {573-583},
year = {2021},
publisher = {University of Guilan},
issn = {2345-394X},
eissn = {2382-9869},
doi = {10.22124/jmm.2021.18991.1625},
abstract = {In this paper, we study the bonus-malus model denoted by $BM_k (n)$. It is an irreducible and aperiodic finite Markov chain but it is not reversible in general. We show that if an irreducible, aperiodic finite Markov chain has a transition matrix whose secondary part is represented by a nonnegative, irreducible and periodic matrix, then we can estimate an explicit upper bound of the coefficient of the leading-order term of the convergence bound. We then show that the $BM_k (n)$ model has the above-mentioned periodicity property. We also determine the characteristic polynomial of its transition matrix. By combining these results with a previously studied one, we obtain essentially complete knowledge on the convergence of the $BM_k (n)$ model in terms of its stationary distribution, the order of convergence, and an upper bound of the coefficient of the convergence bound.},
keywords = {Bonus-malus system,Markov chains,convergence to stationary distribution,the Perron-Frobenius theorem},
url = {https://jmm.guilan.ac.ir/article_4683.html},
eprint = {https://jmm.guilan.ac.ir/article_4683_f88575afeb1b588add985c55d0b88c1d.pdf}
}