Asymptotic behavior of the generalized St. Petersburg sum conditioned on its maximum

Gábor Fukker, László Györfi, Péter Kevei

Published 2013-08-02, updated 2016-02-15Version 2

In this paper, we revisit the classical results on the generalized St. Petersburg sums. We determine the limit distribution of the St. Petersburg sum conditioning on its maximum, and we analyze how the limit depends on the value of the maximum. As an application, we obtain an infinite sum representation of the distribution function of the possible semistable limits. In the representation, each term corresponds to a given maximum, in particular this result explains that the semistable behavior is caused by the typical values of the maximum.