Abstract. For any given sequence $k_n\to\infty $ of integers such that $k_n/n\to0$, we derive a rate at which the distribution functions of suitably centered and normed sums of the $k_n$ largest winnings in $n$ generalized St.Petersburg games merge together uniformly with that sequence of semistable infinitely divisible distribution functions that approximate well the corresponding distribution functions of the total gains in $n$ games. The rate depends on $n$, $k_n$ and the tail parameter of the underlying game.

AMS Subject Classification (1991): 60F05; 60E07, 60G50

Received July 23, 2002. (Registered under 2907/2009.)