Generalized St. Petersburg games revisited

TL;DR

This paper extends classical results on the St. Petersburg game to a generalized setting with a broader payoff distribution, analyzing convergence, truncated versions, and borrowing scenarios.

Contribution

It generalizes existing theorems on the St. Petersburg game, including convergence in distribution and variations involving truncation and borrowing.

Findings

Extended Feller's weak law to the generalized game

Generalized Martin-Löf's 1985-theorem on distribution convergence

Analyzed truncated and borrowing versions in the generalized context

Abstract

The topic of the present paper is a generalized St.\ Petersburg game in which the distribution of the payoff $X$ is given by $P(X=sr^{k-1})=pq^{k-1}$, $k=1,2,\ldots$, where $p+q=1$, and $s,\,r>0$. As for main results, we first extend Feller's classical weak law and Martin-L\"of's 1985-theorem on convergence in distribution along the $2^n$-subsequence. In his 2008-paper Martin-L\"of considers a truncated version of the game and the problem "How much does one gain until 'game over'\,", and a variation where the player can borrow money but has to pay interest on the capital, also for the classical setting. We extend these problems to our more general setting. We close with some additional results and remarks.