Based on a result in a 1987 paper by Bar-Lev and Enis, we show that if $Y_t$ is a driftless subordinator with the additional property that the tail of the Lévy-measure behaves like $-c \cdot \log(x)$ as $x$ tends to zero, then $-t\cdot \log(Y_t)$ tends weakly to a limit having an exponential distribution. We investigate several equivalent conditions to ensure this convergence and present examples of processes that fulfill these conditions. We then prove that one can extend these results to a statement about convergence of processes and show that under the above conditions $-t\cdot \log(Y_{ts})$ tends weakly to what is called an extremal process. Moreover, we present more general results concerning the convergence.

Joint work with Shaul Bar-Lev, Offer Kella and Wolfgang Stadje.