A Maxtrimmed St. Petersburg Game

Let $$S_n$$ S n , $$n\ge 1$$ n ≥ 1 , describe the successive sums of the payoffs in the classical St. Petersburg game. The celebrated Feller weak law states that $$\frac{S_n}{n\log _2 n}\mathop {\rightarrow }\limits ^{p}1$$ S n n log 2 n → p 1 as $$n\rightarrow \infty $$ n → ∞ . It is also known that almost sure convergence fails. However, Csörgő and Simons (Stat Probab Lett 26:65–73, 1996) have shown that almost sure convergence holds for trimmed sums, that is, for $$S_n-\max _{1\le k\le n}X_k$$ S n - max 1 ≤ k ≤ n X k . Since our actual distribution is discrete there may be ties. Our main focus in this paper is on the “maxtrimmed sum”, that is, the sum trimmed by the random number of observations that are equal to the largest one. We prove an analog of Martin-Löf’s (J Appl Probab 22:634–643, 1985) distributional limit theorem for maxtrimmed sums, but also for the simply trimmed ones, as well as for the “total maximum”. In a final section, we interpret these findings in terms of sums of (truncated) Poisson random variables.