Abstract. Even though there are no asymptotic distributions in the usual sense, we show that the distribution functions of the suitably centered and normed cumulative winnings in a full sequence of generalized St.Petersburg games merge together uniformly with completely specified semistable infinitely divisible distribution functions at certain fast rates, depending upon the tail parameter of the game.

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

Received February 12, 2002, and in final form June 25, 2002. (Registered under 2870/2009.)