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On the rate of convergence of the St. Petersburg game. (English) Zbl 1240.60011
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 $\varepsilon >0$, $P(\vert S_{n}^{(n)}-E(S_{n}^{(n)})\vert (n\log_{2}n)^{-1}>\varepsilon )< 2n^{4-(\log\log n)\varepsilon \log_{2}e}$. 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 {\it L. Györfi} and {\it 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.

60E05General theory of probability distributions
60F15Strong limit theorems
60G50Sums of independent random variables; random walks
Full Text: DOI
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