Non-decreasing continuous semi-Markov processes: asymptotics and asymmetry

The authors consider a non-decreasing continuous random process with a family of the first hitting times for levels $x>0$, which form Lévy process with positive increments. Asymptotics of the first three moments of their one-dimensional distributions as t goes to infinity are derived for the case when the Lévy density is $e^{-u}/u^\alpha$ $(1\leq\alpha<2)$.