Extreme-trimmed St. Petersburg games

Let SnSn, n≥1n≥1, describe the successive sums of the payoffs in the classical St. Petersburg game. Feller’s famous weak law, Feller (1945), states that Snnlog2n→p1 as n→∞n→∞. However, almost sure convergence fails, more precisely, lim supn→∞Snnlog2n=+∞ a.s. and lim infn→∞Snnlog2n=1 a.s. as n→∞n→∞. Csörgő and Simons (1996) have shown that almost sure convergence holds for trimmed sums, that is, for Sn−max1≤k≤nXkSn−max1≤k≤nXk and, moreover, that this remains true if the sums are trimmed by an arbitrary fixed number of maximal sums. A predecessor of the present paper was devoted to sums trimmed by the random number of maximal summands. The present paper concerns analogs for the random number of summands equal to the minimum, as well as analogs for joint trimmings.