  Scaled type Markov renewal processes generalize classical renewal processes:
renewal times come from a one parameter family of probability laws and the
sequence of the parameters is the trajectory of an ergodic Markov chain. Our
primary interest here is the asymptotic distribution of the Markovian parameter
at time t \to \infty. The limit, of course, depends on the stationary
distribution of the Markov chain. The results, however, are essentially
different depending on whether the expectations of the renewals are finite or
infinite. If the expectations are uniformly bounded, then we can provide the
limit in general (beyond the class of scaled type processes), where the
expectations of the probability laws in question appear, too. If the means are
infinite, then - by assuming that the renewal times are rescaled versions of a
regularly varying probability law with exponent 0 \leq alpha \leq 1 - it is the
exponent a which emerges in the limits.
