On the concepts of α-recurrence and α-transience for Markov renewal process

AbstractLet I be a denumerable set and let Q = (Qij)i,j∈l be an irreducible semi-Markov kernel. The main results of the paper are: 1.(i) Q is α-recurrent (resp. α-transient, α-positive recurrent, α-null recurrent) if and only if it can be written in the form Qij(dt) = hih−tje−αtQ̂ij(dt), where 0 < hi< ∞ for all i ∈ I, Q̂ is an irreducible, recurrent (resp. transient, positive recurrent, null recurrent) semi-Markov kernel.2.(ii) If Q is α-recurrent, then there is a row vector π = (πi)i∈l and a column vector h = (hi)i∈l, which satisfy π[∫0∞eαtQ(dt)] = π and [∫0∞eαtQ(dt)]h = h.3.(iii) Q is α-positive recurrent if and only if π[∫0∞teαtQ(dt)]h < ∞.Based on the preceding results a Markov renewal limit theorem is proved. We also study the application of our results to Markov processes