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 $({\widehat{\eta }}_s{)}_{s>0}$ 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 $(z\wedge {\widehat{\eta }}_s{)}_{s\ge 0}$ in $\mathcal{D}[0,\mathrm{\infty })$, the space of c`{a}dl`{a}g functions equipped with Skorohod’s $J_1$ metric.