Darling--Erd\H{o}s theorem for L\'evy processes at zero

Abstract

We establish two equivalent versions of the Darling--Erd\H{o}s theorem for L\'evy processes in the domain of attraction of a stable process at zero with index $\alpha\in(0,2)$. In the course of our proof we obtain a number of maximal and exponential inequalities for general L\'evy processes, which should be of separate interest.