Invariance principles for sums of extreme sequential order statistics attracted to Lévy processes

The paper establishes strong convergence results for the joint convergence of sequential order statistics. There exists an explicit construction such that almost sure convergence to extremal processes follows. If a partial sum of rowwise i.i.d. random variables is attracted by a non-Gaussian limit law then the results apply to invariance principles for sums of extreme sequential order statistics which turn out to be almost surely convergent or convergent in probability in D[0,1]. Under certain conditions they converge to the non-Gaussian part of the Lévy process. In addition, we get an approximation of these Lévy processes by a finite number of extremal processes.