Darling-Erdős theorems for normalized sums of i. i. d. variables close to a stable law
Let \(\{S_n\), \(n\geq 1\}\) be partial sums of i.i.d. random variables with mean 0 and finite moment of order three. In the 50's Darling and Erdős proved that \(\max_{k\leq n} k^{-1/2} S_k\), suitably normalized, converges in distribution to the Gumbel distribution. The assumption on third order moments has since then been weakened. In this paper corresponding results are proved in the case when the summands belong to the domain of attraction of a strictly stable law with index \(\alpha\in (0,1)\cup (1,2)\). An important tool is analog for extremes of spectrally negative Ornstein-Uhlenbeck processes.
spectrally negative Ornstein-Uhlenbeck processes
scientific article; zbMATH DE number 1355255
scientific article