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

Abstract

Let I be a denumerable set and let Q = ( Q ij ) 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 Q ij (dt) = h ih −t je −αt Q ̂ ij (dt) , where 0 < h i < ∞ 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 = ( π i ) i ∈ l and a column vector h = ( h i ) i∈ l , which satisfy π[ ∫ 0 ∞ e αtQ (dt)] = π and [ ∫ 0 ∞ e αtQ (dt)]h = h . 3. (iii) Q is α-positive recurrent if and only if π[ ∫ 0 ∞ te αtQ (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.