Let S-n, n >= 1, be the successive sums of the payoffs in the classical St. Petersburg
game. The celebrated Feller weak law states that S-n/(n log(2) n) ->(P) 1 as n ->infinity.
In this paper we review some earlier results of ours and extend some of them as we
consider an asymmetric St. Petersburg game, in which the distribution of the payoff
X is given by P(X = sr(k-1)) = pq(k-1), k = 1, 2,..., where p + q = 1 and s, r > 0.
Two main results are extensions of the Feller weak law and the convergence in distribution
theorem of Martin-Lof (1985). Moreover, it is well known that almost-sure convergence
fails, though Csorgo and Simons (1996) showed that almost-sure convergence holds for
trimmed sums and also for sums trimmed by an arbitrary fixed number of maxima. In
view of the discreteness of the distribution we focus on 'max-trimmed sums', that
is, on the sums trimmed by the random number of observations that are equal to the
largest one, and prove limit theorems for simply trimmed sums, for max-trimmed sums,
as well as for the 'total maximum'. Analogues with respect to the random number of
summands equal to the minimum are also obtained and, finally, for joint trimming.