Reward processes for semi-Markov processes: asymptotic behaviour

The asymptotic behaviour of the cumulative mean of a reward process 𝒵ρ, where the reward function ρ belongs to a rather large class of functions, is obtained. It is proved that E𝒵ρ(t) = C0 + C1t + o(1), t → ∞, where C0 and C1 are fully specified. A section is devoted to the dual process of a semi-Markov process, and a formula is given for the mean of the first passage time from a state i to a state j of the dual process, in terms of the means of passage times of the original process.