We prove Tauberian theorems for random walks with positive drift obeying the central limit theorem. The results include (i) conclusions involving certain averages, relevant to number-theoretic densities and extending results of Diaconis and Stein; (ii) pointwise conclusions, including the classical Borel-Tauber theorem and extending results of Schmaal, Stam and de Vries.
"Tauberian Theorems and the Central Limit Theorem." Ann. Probab. 9 (2) 221 - 231, April, 1981. https://doi.org/10.1214/aop/1176994464