Almost sure limit theorems for the St. Petersburg game
The St. Petersburg game consists of tossing a fair coin until it falls head; if this occurs at the \(k\)th throw, the player receives \(2^k\) ducats. Hence, if \(X\) is the gain at a single trial, we have \(P\{X=2^j\}= 2^{-j}\), \(j=1,2,\dots\) . Let \(X_1, X_2,\dots\) be the player's gains in independent St. Petersburg games and let \(S_n= X_1+ X_2+\cdots+ X_n\). The authors prove that \[ \lim_{n\to\infty} \frac{1}{\log n} \sum_{k=1}^n \frac{1}{k} I\Biggl\{ \frac{S_k}{k}- \text{Log } k\leq x\Biggr\}= G(x) \text{ a.s.} \quad\text{ for all }x\in \mathbb{R}, \] where the limit distribution function \(G\) is explicitly computed. A second a.s. limit distribution theorem involving \(M_n= \max_{i\leq n}X_i\) is also given.
scientific article; zbMATH DE number 1356066