Equivalences in strong limit theorems for renewal counting processes

A number of strong limit theorems for renewal counting processes, e.g. the strong law of large numbers, the Marcinkiewicz-Zygmund law of large numbers or the law of the iterated logarithm, can be derived from their corresponding counterparts for the underlying partial sums. In this paper, it is proved that these strong laws indeed hold simultaneously for both processes. As a byproduct it follows (i) that certain (moment) conditions are necessary and sufficient, and (ii) that the results, in fact, hold for (almost) arbitrary nonnegative summation processes. Renewal processes constructed from random walks with infinite expectation are studied, too, but results are essentially different from the case with linear drift.