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

TL;DR

This paper investigates the asymptotic distribution of generalized St. Petersburg sums conditioned on their maximum, revealing how the maximum influences the limit distribution and providing a sum representation of semistable limits.

Contribution

It determines the limit distribution of the sum conditioned on the maximum and introduces an infinite sum representation of semistable limits based on maximum values.

Findings

Limit distribution depends on the maximum value.

Semistable limits are characterized by an infinite sum representation.

The maximum value influences the asymptotic behavior of the sum.

Abstract

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.