Tauberian Theorems and the Central Limit Theorem

Abstract

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.