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An asymmetric St. Petersburg game with trimming

Part of: Limit theorems Stochastic processes Real functions

Published online by Cambridge University Press:  01 February 2019

Allan Gut  and
Anders Martin-Löf
Allan Gut*
Affiliation:
Uppsala University
Anders Martin-Löf*
Affiliation:
Stockholm University
*
Department of Mathematics, Uppsala University, Box 480, SE-751 06 Uppsala, Sweden. Email address: allan.gut@math.uu.se
Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden. Email address: andersml@math.su.se
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Abstract

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Let Sn,n≥1, be the successive sums of the payoffs in the classical St. Petersburg game. The celebrated Feller weak law states that Sn∕(nlog2n)→ℙ1 as n→∞. In this paper we review some earlier results of ours and extend some of them as we consider an asymmetric St. Petersburg game, in which the distribution of the payoff X is given by ℙ(X=srk-1)=pqk-1,k=1,2,…, where p+q=1 and s,r>0. Two main results are extensions of the Feller weak law and the convergence in distribution theorem of Martin-Löf (1985). Moreover, it is well known that almost-sure convergence fails, though Csörgő and Simons (1996) showed that almost-sure convergence holds for trimmed sums and also for sums trimmed by an arbitrary fixed number of maxima. In view of the discreteness of the distribution we focus on `max-trimmed sums', that is, on the sums trimmed by the random number of observations that are equal to the largest one, and prove limit theorems for simply trimmed sums, for max-trimmed sums, as well as for the `total maximum'. Analogues with respect to the random number of summands equal to the minimum are also obtained and, finally, for joint trimming.

Keywords

St. Petersburg gamesums of independent and identically distributed random variablesFeller's weak law of large numbersconvergence along subsequencesconvergence in distributiontrimmed sum

MSC classification

Primary: 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks
Secondary: 26A12: Rate of growth of functions, orders of infinity, slowly varying functions 60F15: Strong theorems
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue A: Branching and Applied Probability , December 2018 , pp. 115 - 129
Copyright
Copyright © Applied Probability Trust 2018 

References

[1]Adler, A. (1990).Generalized one-sided laws of the iterated logarithm for random variables barely with or without finite mean.J. Theoret. Prob. 3,587–597.Google Scholar
[2]Adler, A. and Rosalsky, A. (1989).On the Chow-Robbins ``fair'' games problem.Bull. Inst. Math. Acad. Sinica 17,211–227.Google Scholar
[3]Berkes, I.,Csáki, E. and Csörgő, S. (1999).Almost sure limit theorems for the St. Petersburg game.Statist. Prob. Lett. 45,23–30.Google Scholar
[4]Berkes, I.,Györfi, L. and Kevei, P. (2017).Tail probabilities of St. Petersburg sums, trimmed sums, and their limit.J. Theoret. Prob. 30,1104–1129.Google Scholar
[5]Chow, Y. S. and Robbins, H. (1961).On sums of independent random variables with infinite moments and ``fair'' games.Proc. Nat. Acad. Sci. USA 47,330–335.Google Scholar
[6]Csörgő, S. (2002).Rates of merge in generalized St. Petersburg games.Acta Sci. Math. (Szeged) 68,815–847.Google Scholar
[7]Csörgő, S. (2007).Merging asymptotic expansions in generalized St. Petersburg games.Acta Sci. Math. (Szeged) 73,297–331.Google Scholar
[8]Csörgő, S. and Simons, G. (1996).A strong law of large numbers for trimmed sums, with applications to generalized St. Petersburg games.Statist. Prob. Lett. 26,65–73.Google Scholar
[9]Feller, W. (1945).Note on the law of large numbers and ``fair'' games.Ann. Math. Statist. 16,301–304.Google Scholar
[10]Feller, W. (1968).An Introduction to Probability Theory and Its Applications, Vol. I,3rd edn.John Wiley,New York.Google Scholar
[11]Feller, W. (1971).An Introduction to Probability Theory and Its Applications, Vol. II,2nd edn.John Wiley,New York.Google Scholar
[12]Gut, A. (2004).An extension of the Kolmogorov-Feller weak law of large numbers with an application to the St. Petersburg game.J. Theoret. Prob. 17,769–779.Google Scholar
[13]Gut, A. (2010).Limit theorems for a generalized St. Petersburg game.J. Appl. Prob. 47,752–760. (Correction: available at http://www.math.uu.se/∼allan/86correction.pdf.)Google Scholar
[14]Gut, A. (2013).Probability: A Graduate Course,2nd edn.Springer,New York.Google Scholar
[15]Gut, A. and Martin-Löf, A. (2015).Extreme-trimmed St. Petersburg games.Statist. Prob. Lett. 96,341–345.Google Scholar
[16]Gut, A. and Martin-Löf, A. (2015). Generalized St. Petersburg games revisited. Preprint. Available at https://arxiv.org/abs/1506.09015v1.Google Scholar
[17]Gut, A. and Martin-Löf, A. (2016).A maxtrimmed St. Petersburg game.J. Theoret. Prob. 29,277–291.Google Scholar
[18]Martin-Löf, A. (1985).A limit theorem which clarifies the `Petersburg paradox'.J. Appl. Prob. 22,634–643.Google Scholar