Reducible Markov Modulation, Pole Order, and Tail Behavior in Random Growth Models

This is a follow up paper of Beare & Toda (2022) and Beare, Seo, & Toda (2022), where we characterize the tail behavior of stopped Markov additive processes in discrete- and continuous-time, respectively. In these papers, we assumed that the transition probability matrix of the Markov chain is irreducible. While writing the latter paper, in 2021 I obtained a complete characterization of the tail behavior even if the transition probability matrix is reducible. However, the argument was complicated, so we saved the result for another paper. Subsequently, Brendan discovered a connection to the Rothblum index theorem.