Karamata Method and the Maximum of Sums of Independent Random Variables

We revisit the classical problem of convergence of the maximum of cumulative sums of IID random variables by introducing ideas from the Karamata's celebrated proof of the Hardy-Littlewood Tauberian Theorem [J. Karamata, Über die Hardy-Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes, Math. Z. 32 (1930), 319-320]. As a main result we present an extension of the Darling-Heyde convergence device, and use it to provide a direct analytic proof of convergence in distribution of normalized cumulative sums of IID random variables. The introduction of Karamata's ideas makes the proof self-contained and relying only on basic regular variation results