  The topic of the present paper is a generalized St.\ Petersburg game in which
the distribution of the payoff $X$ is given by $P(X=sr^{k-1})=pq^{k-1}$,
$k=1,2,\ldots$, where $p+q=1$, and $s,\,r>0$. As for main results, we first
extend Feller's classical weak law and Martin-L\"of's 1985-theorem on
convergence in distribution along the $2^n$-subsequence. In his 2008-paper
Martin-L\"of considers a truncated version of the game and the problem "How
much does one gain until 'game over'\,", and a variation where the player can
borrow money but has to pay interest on the capital, also for the classical
setting. We extend these problems to our more general setting. We close with
some additional results and remarks.

Probability