Darling-Erdős theorems for normalized sums of i.i.d. variables close to a stable law

Abstract

Let $\xi, \xi_1, \dots$ be i.i.d. real-valued random variables and $S_n = \xi_1 + \dots + \xi_n$. In the case when the distribution of $\xi$ is close to a stable $(\alpha)$ law for some $\alpha \epsilon (0, 1) \bigcup (1, 2)$, we investigate the asymptotic behavior in distribution of the maximum of normalized sums, $\max_{k=1,\dots,n} k^{-1/\alpha}S_k$. This completes the Darling-Erdös limit theorem for the case $\alpha = 2$.