Large deviations for compound Markov renewal processes with dependent jump sizes and jump waiting times

Abstract

In [17] the author considered a compound Markov renewal process $(\widetilde{S}_{N_t})$ where $((J_n,S_n))$ and $((\widetilde{J}_n,\widetilde{S}_n))$ are suitable independent Markov additive processes such that $(S_n-S_{n-1})$ are positive random variables, and $N_t=\sum _{n\geq 1}1_{S_n\leq t}$. In this paper we present the analogous results for a more general situation where we consider a unique Markov additive process $((J_n,Z_n))$ in place of $((J_n,S_n))$ and $((\widetilde{J}_n,\widetilde{S}_n))$, and $Z_n=(\widetilde{S}_n,S_n)$. Some further results are also presented; in particular we relate in terms of large deviations the sequence $((\widetilde{S} _n,S_n))$ and the process $((\widetilde{S}_{N_t},N_t))$.