On the rate of convergence of the St. Petersburg game
Let \(P(X_{n}=2^{k})=2^{- k}\) with independent \(X_{n}\), \(X_{n}^{(c)}=\min(X_{n},c)\), \(S_{n}^{(c)}=\sum_{k=1}^{n}X_{k}^{(c)}\). \(\gamma_{n} {\rightarrow}^{(c)} \gamma\) means either \(\gamma \in (1/2,1)\), \(\gamma_{n}\rightarrow \gamma\) or \(\gamma =1\) and \(\gamma_{n}\) has no other limit points than \(1\) and \(1/2\); \(\{x\}\), \([x]\) denote the fractional and the integer parts of \(x\). The authors prove the following theorems. 1. For \(\varepsilon >0\), \(P(| S_{n}^{(n)}-E(S_{n}^{(n)})| (n\log_{2}n)^{-1}>\varepsilon )< 2n^{4-(\log\log n)\varepsilon \log_{2}e}\). 2. \(n^{-1}\sum_{i=1}^{n}(X_{i}-n)^{+}\), \(n=n_{k}\), converges in distribution for \(k\rightarrow \infty\) to some nondegenerate limit if and only if \(n_k/2^{[\log_2n_k]} {\rightarrow}^{(c)} \gamma\), and the limit has the characteristic function \(\exp(\int_{_{0}}^{^{\infty }}(e^{itx}-1-itx(1+x^{2})^{-1})d(-2^{\{\log_{2}[\gamma (x+1)]\}}(x+1)^{-1}))\). 3. Same for \(n_{k}^{-1}S_{n_{k}}^{(n_{k})}-\log_{2}n_{k}\) with \(-x^{-1} 2^{\{\log_{2}(\gamma x)\}}\) for \(x<1\), \(0\) otherwise, under \(d\). 4. \((VarS_{n}^{c_{n}})^{-1/2}(S_{n}^{c_{n}}-E(S_{n}^{c_{n}}))\) tends in distribution to the standard normal one if and only if \(c_{n}/n\rightarrow 0\). 5. \((0.16+o(1))/\log_{2}n\leq E(\log_{2}(S_{n}/(n\log_{2}n)))-\log_{2}\log_{2}n/(\log 2)(\log_{2}n)\leq (2.52+o(1))/\log_{2} n\). 6. \(E((\log_{2}(S_{n}/ (n\log_{2}n)) )^{2}) =O(1/\log n)\). Theorem 5 appears in a paragraph entitled ``Growth rate of sequential St. Petersburg portfolio games'' in which some results of \textit{L. Györfi} and \textit{P. Kevei} [Algorithmic learning theory. Proceedings. Berlin: Springer. Lecture Notes in Computer Science 5809. Lecture Notes in Artificial Intelligence, 83--96 (2009; Zbl 1262.91047)] are presented. The paper finishes by showing histograms of some \(\log_{2}S_{n}\), \(\log_{2}S_{n}^{(c)}\) and by proving that \(\log_{2}S_{n}\) is not asymptotically normal.
scientific article; zbMATH DE number 5990581
scientific article