The dimension of the St. Petersburg game

Abstract

Let $S_n$ be the total gain in $n$ repeated St.\ Petersburg games. It is known that $n^{-1}(S_n-n{\log }_{2}n)$ converges in distribution to a random element $Y(t)$ along subsequences of the form $k(n)=2^{p(n)}t(n)$ with $p(n)=\lceil {\log }_{2}k(n)\rceil \to \infty$ and $t(n)\to t\in [\frac{1}{2},1]$. 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.

References in corpus (1)

• Hausdorff dimension of operator semistable Lévy processes