Invariance principles for sums of extreme sequential order statistics attracted to Levy 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 su m of rowwise i.i.d. random variables is attracted by a non-Gaussian limit l aw then the results apply to invariance principles for sums of extreme sequ ential order statistics which turn out to be almost surely convergent or co nvergent in probability in D[0, 1]. Under certain conditions they converge to the non-Gaussian part of the Levy process. In addition, we get an approx imation of these Levy processes by a finite number of extremal processes. ( C) 2000 Elsevier Science B.V. All rights reserved.