The topic of the present paper is a generalized St Petersburg game in which the distribution of the payoff X is given by P(X =
sr
(k-1)/α) = pq
k-1,
k = 1, 2,…, where p + q = 1, s = 1 / p,
r = 1 / q, and 0 < α ≤ 1. For the case in which α = 1, we extend Feller's classical weak law and Martin-Löf's theorem on convergence in distribution along the 2
n
-subsequence. The analog for 0 < α < 1 turns out to converge in distribution to an asymmetric stable law with index α. Finally, some limit theorems for polynomial and geometric size total gains, as well as for extremes, are given.