Maller, RossSchindler, Tanja2021-02-050304-4149http://hdl.handle.net/1885/222049We consider subordinators in the domain of attraction at 0 of a stable subordinator (where ); thus, with the property that , the tail function of the canonical measure of , is regularly varying of index as . We also analyse the boundary case, , when is slowly varying at 0. When , we show that converges in distribution, as , to the random variable . This latter random variable, as a function of , converges in distribution as to the inverse of an exponential random variable. We prove these convergences, also generalised to functional versions (convergence in ), and to trimmed versions, whereby a fixed number of its largest jumps up to a specified time are subtracted from the process. The case produces convergence to an extremal process constructed from ordered jumps of a Cauchy subordinator. Our results generalise random walk and stable process results of Darling, Cressie, Kasahara, Kotani and Watanabe.application/pdfen-AU© 2018 Elsevier B.VTrimmed subordinatorLévy processMaximal jump processFunctional convergenceRegular variationExtremal processCauchy process201910.1016/j.spa.2018.11.0162020-11-02