Merging to Semistable Laws

We show that, despite the lack of asymptotic distributions in the usual sense, the distribution functions of suitably centered and normalized partial sums of independent and identically distributed random variables from the domain of geometric partial attraction of a semistable law merge together uniformly with a family of semistable distribution functions. More generally, even the corresponding, possibly lightly trimmed sums merge. Analogous merge results hold also for sample extremes. The main result is illustrated on generalized St. Petersburg games.