In this paper, we prove an analogue of the classical renewal theorem for the case where there is no drift. Our proof depends on a uniform version of Spitzer's well-known theorem on ladder epochs and ladder variables, and we obtain this uniform result by using uniform Tauberian theorems. Some further applications of these uniform Tauberian theorems to other problems in renewal theory and first passage times are also given.
"Uniform Tauberian Theorems and their Applications to Renewal Theory and First Passage Problems." Ann. Probab. 4 (4) 628 - 643, August, 1976. https://doi.org/10.1214/aop/1176996032