University of Guilan Journal of Mathematical Modeling 2345-394X 9 4 2021 12 01 On the spectral properties and convergence of the bonus-malus Markov chain model 573 583 4683 10.22124/jmm.2021.18991.1625 EN Kenichi Hirose 10-17 Moto-machi, Ono City, Fukui 912-0081, Japan Journal Article 2021 02 22 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.<br /><br /> https://jmm.guilan.ac.ir/article_4683_f88575afeb1b588add985c55d0b88c1d.pdf