  We consider random variables observed at arrival times of a renewal process,
which possibly depends on those observations and has regularly varying steps
with infinite mean. Due to the dependence and heavy tailed steps, the limiting
behavior of extreme observations until a given time $t$ tends to be rather
involved. We describe this asymptotics and generalize several partial results
which appeared in this setting. In contrast to the earlier studies, our
analysis is based in the point processes theory. The theory is applied to
determine the asymptotic distribution of maximal excursions and sojourn times
for continuous time random walks.
