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Limit Theorems for a Generalized ST Petersburg Game

Part of: Limit theorems Stochastic processes Real functions

Published online by Cambridge University Press:  14 July 2016

Allan Gut
Allan Gut*
Affiliation:
Uppsala University
*
∗Postal address: Department of Mathematics, Uppsala University, Box 480, SE-751 06 Uppsala, Sweden. Email address: [email protected]
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Abstract

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The topic of the present paper is a generalized St Petersburg game in which the distribution of the payoff X is given by P(X = sr(k-1)/α) = pqk-1, k = 1, 2,…, where p + q = 1, s = 1 / p, r = 1 / q, and 0 < α ≤ 1. For the case in which α = 1, we extend Feller's classical weak law and Martin-Löf's theorem on convergence in distribution along the 2n-subsequence. The analog for 0 < α < 1 turns out to converge in distribution to an asymmetric stable law with index α. Finally, some limit theorems for polynomial and geometric size total gains, as well as for extremes, are given.

Keywords

St Petersburg gamesums of i.i.d. random variablesFeller's weak law of large numbersdomains of attractionconvergence along subsequencesextremesstable lawslow variationregular variation

MSC classification

Secondary: 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks 26A12: Rate of growth of functions, orders of infinity, slowly varying functions
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 3 , September 2010 , pp. 752 - 760
Copyright
Copyright © Applied Probability Trust 2010 

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] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.Google Scholar
[4] 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
[5] 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
[6] Feller, W. (1945). Note on the law of large numbers and ‘fair’ games. Ann. Math. Statist. 16, 301–304.Google Scholar
[7] Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol 1, 3rd edn. John Wiley, New York.Google Scholar
[8] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol 2, 2nd edn. JohnWiley, New York.Google Scholar
[9] 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
[10] Gut, A. (2007). Probability: A Graduate Course. Springer, New York.Google Scholar
[11] Hu, Y. and Nyrhinen, H. (2004). Large deviations view points for heavy-tailed random walks. J. Theoret. Prob. 17, 761–768.Google Scholar
[12] Kevei, P. (2009). Linear combinations of i.i.d. random variables from the domain of geometric partial attraction of a semistable law. , Szeged University (in Hungarian).Google Scholar
[13] Martin-Löf, A. (1985). A limit theorem which clarifies the ‘Petersburg paradox’. J. Appl. Prob. 22, 634–643.Google Scholar
[14] Stoica, G. (2008). Large gains in the St. Petersburg game. C. R. Acad. Sci. Paris 346, 563–566.Google Scholar