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On the rate of convergence of the St. Petersburg game. (English) Zbl 1240.60011

Let $P\left({X}_{n}={2}^{k}\right)={2}^{-k}$ with independent ${X}_{n}$, ${X}_{n}^{\left(c\right)}=min\left({X}_{n},c\right)$, ${S}_{n}^{\left(c\right)}={\sum }_{k=1}^{n}{X}_{k}^{\left(c\right)}$. ${\gamma }_{n}{\to }^{\left(c\right)}\gamma$ means either $\gamma \in \left(1/2,1\right)$, ${\gamma }_{n}\to \gamma$ or $\gamma =1$ and ${\gamma }_{n}$ has no other limit points than 1 and $1/2$; $\left\{x\right\}$, $\left[x\right]$ denote the fractional and the integer parts of $x$. The authors prove the following theorems.

1. For $\epsilon >0$, $P\left(|{S}_{n}^{\left(n\right)}-E\left({S}_{n}^{\left(n\right)}\right)|{\left(n{log}_{2}n\right)}^{-1}>\epsilon \right)<2{n}^{4-\left(loglogn\right)\epsilon {log}_{2}e}$.

2. ${n}^{-1}{\sum }_{i=1}^{n}{\left({X}_{i}-n\right)}^{+}$, $n={n}_{k}$, converges in distribution for $k\to \infty$ to some nondegenerate limit if and only if ${n}_{k}/{2}^{\left[{log}_{2}{n}_{k}\right]}{\to }^{\left(c\right)}\gamma$, and the limit has the characteristic function $exp\left({\int }_{{}_{0}}^{{}^{\infty }}\left({e}^{itx}-1-itx{\left(1+{x}^{2}\right)}^{-1}\right)d\left(-{2}^{\left\{{log}_{2}\left[\gamma \left(x+1\right)\right]\right\}}{\left(x+1\right)}^{-1}\right)\right)$.

3. Same for ${n}_{k}^{-1}{S}_{{n}_{k}}^{\left({n}_{k}\right)}-{log}_{2}{n}_{k}$ with $-{x}^{-1}{2}^{\left\{{log}_{2}\left(\gamma x\right)\right\}}$ for $x<1$, 0 otherwise, under $d$.

4. ${\left(Var{S}_{n}^{{c}_{n}}\right)}^{-1/2}\left({S}_{n}^{{c}_{n}}-E\left({S}_{n}^{{c}_{n}}\right)\right)$ tends in distribution to the standard normal one if and only if ${c}_{n}/n\to 0$.

5. $\left(0·16+o\left(1\right)\right)/{log}_{2}n\le E\left({log}_{2}\left({S}_{n}/\left(n{log}_{2}n\right)\right)\right)-{log}_{2}{log}_{2}n/\left(log2\right)\left({log}_{2}n\right)\le \left(2·52+o\left(1\right)\right)/{log}_{2}n$.

6. $E\left({\left({log}_{2}\left({S}_{n}/\left(n{log}_{2}n\right)\right)\right)}^{2}\right)=O\left(1/logn\right)$.

Theorem 5 appears in a paragraph entitled “Growth rate of sequential St. Petersburg portfolio games” in which some results of L. Györfi and 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 ${log}_{2}{S}_{n}$, ${log}_{2}{S}_{n}^{\left(c\right)}$ and by proving that ${log}_{2}{S}_{n}$ is not asymptotically normal.

##### MSC:
 60E05 General theory of probability distributions 60F15 Strong limit theorems 60G50 Sums of independent random variables; random walks
##### References:
 [1] A. Adler, Generalized one-sided laws of iterated logarithm for random variables barely with or without finite mean, J. Theoret. Probab., 3 (1990), 587–597. · Zbl 0712.60029 · doi:10.1007/BF01046098 [2] D. Bernoulli, Exposition of a new theory on the measurement of risk, Econometrica, 22 (1954), 22–36 (Originally published in 1738; translated by L. Sommer). · Zbl 0055.12004 · doi:10.2307/1909829 [3] L. Breiman, Optimal gambling systems for favorable games, Proc. Fourth Berkeley Symp. Math. Statist. Prob. 1, Univ. California Press, Berkeley, 1961, 65–78. [4] Y. S. Chow and H. Robbins, On sums of independent random variables with infinite moments and ”fair” games, Proc. Nat. Acad. Sci. USA, 47 (1961), 330–335. · Zbl 0099.35103 · doi:10.1073/pnas.47.3.330 [5] S. Csörgo, Rates of merge in generalized St. Petersburg games, Acta Sci. Math. (Szeged), 68 (2002), 815–847. [6] S. Csörgo, A szentpétervári paradoxon, Polygon, 5 (1995), 19–79, in Hungarian. [7] S. Csörgo, Merging asymptotic expansions in generalized St. Petersburg games, Acta Sci. Math. (Szeged), 73 (2007), 297–331. [8] S. Csörgo and R. Dodunekova, Limit theorems for the Petersburg game, Sums, Trimmed Sums and Extremes, Progress in Probability 23 (eds. M. G. Hahn, D. M. Mason and D. C. Weiner), Birkhäuser, Boston, 1991, 285–315. [9] S. Csörgo and P. Kevei, Merging asymptotic expansions for cooperative gamblers in generalized St. Petersburg games, Acta Math. Hungar., 121 (2008), 119–156. · Zbl 1199.60052 · doi:10.1007/s10474-008-7193-8 [10] S. Csörgo and Z. Megyesi, Merging to semistable laws, Theory Probab. Appl., 47 (2002), 17–33. · Zbl 1043.60014 · doi:10.1137/S0040585X97979470 [11] S. Csörgo and G. Simons, A strong law of large numbers for trimmed sums, with applications to generalized St. Petersburg games, Statist. Probab. Lett., 26 (1996), 65–73. · Zbl 0859.60030 · doi:10.1016/0167-7152(94)00253-3 [12] W. Feller, Note on the law of large numbers and ”fair” games, Ann. Math. Statistics, 16 (1945), 301–304. · Zbl 0060.28701 · doi:10.1214/aoms/1177731094 [13] B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Reading, Massachusetts, 1954. [14] L. Györfi and P. Kevei, St. Petersburg portfolio games, Proceedings of Algorithmic Learning Theory 2009 (eds. R. Gavaldà et al.), Lecture Notes in Artificial Intelligence 5809, Springer, 2009, 83–96. [15] J. L. Kelly, A new interpretation of information rate, Bell System Tech. J., 35 (1956), 917–926. · doi:10.1002/j.1538-7305.1956.tb03809.x [16] A. Martin-löf, A limit theorem which clarifies the ”Petersburg paradox”, J. Appl. Probab., 22 (1985), 634–643. · Zbl 0574.60032 · doi:10.2307/3213866