Abstract. Merging asymptotic expansions are established for the distribution functions of suitably centered and normed cumulative winnings in a full sequence of generalized St.$ $Petersburg games. These expansions are given in terms of suitably chosen members from the classes of subsequential semistable infinitely divisible asymptotic distribution functions and certain derivatives of these functions, where the classes themselves are determined by the two parameters of the game. Depending upon the most interesting cases of the tail parameter, which include the classical St.$ $Petersburg game, the expansions yield best possible rates of uniform merge with the selected semistable distribution functions.

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

Received January 18, 2007, and in final form May 16, 2007. (Registered under 5968/2009.)