University of GuilanJournal of Mathematical Modeling2345-394X9420211201On the spectral properties and convergence of the bonus-malus Markov chain model573583468310.22124/jmm.2021.18991.1625ENKenichiHirose10-17 Moto-machi, Ono City, Fukui 912-0081, JapanJournal Article20210222In 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