Tail Probabilities of St. Petersburg Sums, Trimmed Sums, and Their Limit

We provide exact asymptotics for the tail probabilities P{Sn>x} and P{Sn−X∗n>x} as x→∞, for fix n, where Sn and X∗n is the partial sum and partial maximum of i.i.d. St. Pe- tersburg random variables. We show that while the order of the tail of the sum Sn is x−1 , the order of the tail of the trimmed sum Sn−X∗n is x−2 . In particular, we prove that al- though the St. Petersburg distribution is only O-subexponential, the subexponential property almost holds. We also provide an infinite series representation of the distribution function of the limiting distribution of the trimmed sum, and analyze its tail behavior.