  We study null recurrent renewal Markov chains with renewal distribution in
the domain of geometric partial attraction of a semistable law. Using the
classical procedure of inversion, we derive a limit theorem similar to the
Darling-Kac law along subsequences and obtain some interesting properties of
the limit distribution. Also in this context, we obtain a Karamata type theorem
along subsequences for positive operators. In both results, we identify the
allowed class of subsequences. We provide several examples of nontrivial
infinite measure preserving systems to which these results apply.

Dynamical Systems;Probability