The classical Doeblin-Gnedenko conditions characterizing the domain of attraction of a non-Gaussian stable law have been proved to be sufficient for completely non-Gaussian stable continuous convolution semigroups (c.c.s. for short) on simply connected nilpotent Lie groups by Carnal [2]. His method was a translation of the extreme-value-theoretic approach due to Le Page, Woodroofe, Zinn [11]. In the present note, we give a generalization of the classical proof for the sufficiency of these conditions. Together with Neuenschwander
[13] this yields the fact that in case every eigenvalue of the square matrix A has real part strictly greater than 1/2, then for a completely non-Gaussian {tA}t>0-stable (in the not necessarily strict sense) semigroup on simply connected nilpotent Lie groups, the analogue of the Doeblin-Gnedenko conditions in fact characterize the non-strict {tA}t>0-domain of attraction. We take this opportunity to state a new proof of Siebert's Convergence Theorem for c.c.s. and their generating distributions for the special case of simply connected nilpotent Lie groups and related results.