Strong limit theorems for increments of renewal processes

We study the almost surely behavior of increments of renewal processes. We derive a universal form of norming functions in strong limit theorems for increments of such processes. This unifies the following well known theorems for increments of renewal processes: the strong law of large numbers, the Erdős–Rényi law, the Csörgő-Révész law and the law of the iterated logarithm. New results are obtained for processes with distributions of renewal times from domains of attraction of a normal law and completely asymmetric stable laws with index $\alpha\in(1,2)$.