The dimension of the St. Petersburg game

TL;DR

This paper investigates the fractal dimensions of the stochastic process derived from the scaled total gains in repeated St. Petersburg games, revealing detailed geometric properties of its sample paths.

Contribution

It determines the Hausdorff and box-counting dimensions of the process's range and graph, extending understanding of its fractal structure.

Findings

Hausdorff dimension of the range and graph is explicitly calculated

Box-counting dimension results are established for the process

Comparison with deterministic sequences highlights differences in fractal properties

Abstract

Let $S_n$ be the total gain in $n$ repeated St.\ Petersburg games. It is known that $n^{-1}(S_n-n\log_2n)$ converges in distribution to a random element $Y(t)$ along subsequences of the form $k(n)=2^{p(n)}t(n)$ with $p(n)=\lceil\log_2k(n)\rceil\to\infty$ and $t(n)\to t\in[\frac12,1]$. We determine the Hausdorff and box-counting dimension of the range and the graph for almost all sample paths of the stochastic process $\{Y(t)\}_{t\in[1/2,1]}$. The results are compared to the fractal dimension of the corresponding limiting objects when gains are given by a deterministic sequence initiated by Hugo Steinhaus.