Weak convergence of subordinators to extremal processes

For certain subordinators (Xt)t≥0 it is shown that the process (−tlogXts)s>0 tends to an extremal process (η̂s)s>0 in the sense of convergence of the finite dimensional distributions. Additionally it is also shown that (z∧(−tlogXts))s≥0 converges weakly to (z∧η̂s)s≥0 in D[0,∞), the space of càdlàg functions equipped with Skorohod's J1 metric.