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

## TL;DR

This paper extends the Darling--Erdős theorem to Lévy processes near zero, providing two equivalent formulations and new inequalities, advancing understanding of their asymptotic behavior.

## Contribution

It introduces two equivalent versions of the Darling--Erdős theorem for Lévy processes at zero and derives new maximal and exponential inequalities.

## Key findings

- Two equivalent formulations of the Darling--Erdős theorem for Lévy processes at zero.
- New maximal inequalities for general Lévy processes.
- New exponential inequalities for general Lévy processes.

## 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.