Abstract. We give a probabilistic representation of all possible limiting distributions of sums of independent identically distributed random variables along subsequences of ${\msbm N}$ satisfying a geometrical growth condition and describe the distributions partially attracted to them. We investigate the structure of these domains of partial attraction and also discuss the effect of light trimming.

AMS Subject Classification (1991): 60F05, 60E07

Keyword(s): Semistable laws, domains of partial attraction, lightly trimmed sums

Received April 23, 1999, and in revised form August 30, 1999. (Registered under 2742/2009.)