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,...

This, somewhat unusual collection of problems in Measure-Theoretic Probability and Stochastic Analysis, should have been more appropriately called: ``The Problems I Like". For the material borrowed from the existing literature, full references and occasional historical remarks are provided....

In this work we provide generalization and unification of several moment formulae: the Lee Moment Formula of Lee (2004), Log Moment Formula of Raval and Jacquier (2021), and the modified Piterbarg conjecture of Gulisashvili (2012). We approach the problem via investigating asymptotic behavior of...