On the concepts of [alpha]-recurrence and [alpha]-transience for Markov renewal process

Let I be a denumerable set and let Q = (Qij)i,j[set membership, variant]l be an irreducible semi-Markov kernel. The main results of the paper are: 1. (i) Q is [alpha]-recurrent (resp. [alpha]-transient, [alpha]-positive recurrent, [alpha]-null recurrent) if and only if it can be written in the form 73, where 0 < hi< [infinity] for all i [set membership, variant] I, Q is an irreducible, recurrent (resp. transient, positive recurrent, null recurrent) semi-Markov kernel. 2. (ii) If Q is [alpha]-recurrent, then there is a row vector [pi] = ([pi]i)i[set membership, variant]l and a column vector h = (hi)i[set membership, variant]l, which satisfy 40 and 29. 3. (iii) Q is [alpha]-positive recurrent if and only if 52. Based on the preceding results a Markov renewal limit theorem is proved. We also study the application of our results to Markov processes.