The accuracy of merging approximation in generalized St. Petersburg games

TL;DR

This paper develops detailed merging asymptotic expansions for the distribution functions and probabilities in generalized St. Petersburg games, extending previous short expansions and providing bounds based on the tail parameter.

Contribution

It introduces long merging asymptotic expansions for generalized St. Petersburg games using subsequential semistable infinitely divisible distributions, extending prior short expansions.

Findings

Derived asymptotic expansions of arbitrary length.

Provided uniform and nonuniform bounds.

Extended previous results to more general cases.

Abstract

Merging asymptotic expansions of arbitrary length are established for the distribution functions and for the probabilities of suitably centered and normalized cumulative winnings in a full sequence of generalized St. Petersburg games, extending the short expansions due to Cs\"org\H{o}, S., Merging asymptotic expansions in generalized St. Petersburg games, \textit{Acta Sci. Math. (Szeged)} \textbf{73} 297--331, 2007. These expansions are given in terms of suitably chosen members from the classes of subsequential semistable infinitely divisible asymptotic distribution functions and certain derivatives of these functions. The length of the expansion depends upon the tail parameter. Both uniform and nonuniform bounds are presented.