The dimension of the St. Petersburg game

摘要

Let $S_n$ be the total gain in $n$ repeated St.\ Petersburg games. It isknown 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 $t(n)\to t\in[\frac12,1]$. Wedetermine the Hausdorff and box-counting dimension of the range and the graphfor 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 limitingobjects when gains are given by a deterministic sequence initiated by HugoSteinhaus.