Weak Convergence of Subordinators to Extremal Processes

For certain subordinators $(X_t)_{t\ge 0}$ it is shown that the process $(-t\log X_{ts})_{s>0}$ tends to an extremal process $(\hat\eta_s)_{s>0}$ in the sense of convergence of the finite dimensional distributions. Additionally it is also shown that $(z\wedge(-t\log X_{ts}))_{s\ge 0}$ converges weakly to $(z\wedge\hat\eta_s)_{s\ge0}$ in $\mathcal{D}[0,\infty)$, the space of c\`{a}dl\`{a}g functions equipped with Skorohod's $J_1$ metric.