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

In [17] the author considered a compound Markov renewal process ((S) over tilde (Nt)) where ((J(n), S-n)) and (((J) over tilde (n), (S) over tilde (n))) are suitable independent Markov additive processes such that (S-n - Sn-1) are positive random variables, and N-t = Sigma(n >= 1) > 1(Sn <= 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 (((J) over tilde (n), (S) over tilde (n)))and Z(n) = ((S) over tilde (n), S-n). Some further results are also presented; in particular we relate in terms of large deviations the sequence (((S) over tilde (n), S-n)) and the process (((S) over tilde (Nt), N-t)).