A strong invariance principle for the logarithmic average of sample maxima

Given an extremal-Λ process {YΛ(t), t>0}, the transformed process {U(s)=YΛ(es)−s, −∞<s<∞} is a stationary strong Markov process. We prove an almost sure invariance principle for the process {∫0tf(Us) ds, t⩾0}. By an approximation this yields an almost sure invariance principle for the logarithmic average of normed sample maxima, which have been investigated recently in various papers. With this invariance principle, we can also get various results on the behavior of sums of minima of a sequence of random variables.