%0 Journal Article
%T On the spectral properties and convergence of the bonus-malus Markov chain model
%J Journal of Mathematical Modeling
%I University of Guilan
%Z 2345-394X
%A Hirose, Kenichi
%D 2021
%\ 12/01/2021
%V 9
%N 4
%P 573-583
%! On the spectral properties and convergence of the bonus-malus Markov chain model
%K Bonus-malus system
%K Markov chains
%K convergence to stationary distribution
%K the Perron-Frobenius theorem
%R 10.22124/jmm.2021.18991.1625
%X 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.
%U https://jmm.guilan.ac.ir/article_4683_f88575afeb1b588add985c55d0b88c1d.pdf