Weak convergence of subordinators to extremal processes

For certain subordinators ( X t ) t ≥ 0 it is shown that the process ( − t log X t s ) s > 0 tends to an extremal process ( η ̂ s ) s > 0 in the sense of convergence of the finite dimensional distributions. Additionally it is also shown that ( z ∧ ( − t log X t s ) ) s ≥ 0 converges weakly to ( z ∧ η ̂ s ) s ≥ 0 in D [ 0 , ∞ ) , the space of càdlàg functions equipped with Skorohod’s J 1 metric.