Geodesic
Strong limit theorems for increments of renewal processes
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 6, Tome 298 (2003), pp. 208-225

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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 $\alpha\in(1,2)$. 
@article{ZNSL_2003_298_a13,
     author = {A. N. Frolov},
     title = {Strong limit theorems for increments of renewal processes},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {208--225},
     publisher = {mathdoc},
     volume = {298},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_298_a13/}
}
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A. N. Frolov. Strong limit theorems for increments of renewal processes. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 6, Tome 298 (2003), pp. 208-225. http://geodesic.mathdoc.fr/item/ZNSL_2003_298_a13/