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Merging asymptotic expansions for cooperative gamblers in generalized St. Petersburg games. (English) Zbl 1199.60052
Merging asymptotic expansions are established for the distribution functions of suitably centered and normed linear combinations of winnings in a full sequence of generalized St. Petersburg games, where a linear combination is viewed as the share of any one of n cooperative gamblers, who play with a pooling strategy. The expansions are given in terms of Fourier-Stieltjes transforms and are constructed from suitably chosen members of the classes of subsequential semistable infinitely divisible asymptotic distributions for the total winnings of the n players and from their pooling strategy, where the classes themselves are determined by the two parameters of game. For all values of the tail parameter, the expansions yield best possible rates of uniform merge. Surprisingly, it turns out that for a subclass of strategies, not containing the averaging uniform strategy, the merging approximations reduce to asymptotic expansions of the usual type, derived from a proper limiting distribution. The Fourier-Stieltjes transforms are shown to be numerically invertible in general and it is also demonstrated that the merging expansions provide excellent approximations even for very small n.
60F05Central limit and other weak theorems
60E07Infinitely divisible distributions; stable distributions
60G40Stopping times; optimal stopping problems; gambling theory
60G50Sums of independent random variables; random walks
91A60Probabilistic games; gambling
[1]S. Csörgo, Rates of merge in generalized St. Petersburg games, Acta Sci. Math. (Szeged), 68 (2002), 815–847.
[2]S. Csörgo, A probabilistic proof of Kruglov’s theorem on the tails of infinitely divisible distributions, Acta Sci. Math. (Szeged), 71 (2005), 405–415.
[3]S. Csörgo, Merging asymptotic expansions in generalized St. Petersburg games, Acta Sci. Math. (Szeged), 73 (2007), 297–331.
[4]S. Csörgo, Fourier analysis of semistable distributions, Acta Appl. Math., 96 (2007), 159–175. · Zbl 1117.60015 · doi:10.1007/s10440-007-9111-4
[5]S. Csörgo and R. Dodunekova, Limit theorems for the Petersburg game, in: Sums, Trimmed Sums and Extremes (M. G. Hahn, D. M. Mason and D. C. Weiner, eds.), Progress in Probability 23, Birkhäuser (Boston, 1991), pp. 285–315.
[6]S. Csörgo and Z. Megyesi, Merging to semistable laws, Teor. Veroyatn. Primen., 47 (2002), 90–109. [Theory Probab. Appl., 47 (2002), 17–33.]
[7]S. Csörgo and G. Simons, The two-Paul paradox and the comparison of infinite expectations, in: Limit Theorems in Probability and Statistics (Eds. I. Berkes, E. Csáki and M. Csörgo), Vol. I, János Bolyai Mathematical Society (Budapest, 2002), pp. 427–455.
[8]S. Csörgo and G. Simons, Laws of large numbers for cooperative St. Petersburg gamblers, Period. Math. Hungar., 50 (2005), 99–115. · Zbl 1113.60026 · doi:10.1007/s10998-005-0005-9
[9]S. Csörgo and G. Simons, Pooling strategies for St. Petersburg gamblers, Bernoulli, 12 (2006), 971–1002. · Zbl 1130.91018 · doi:10.3150/bj/1165269147
[10]J. Gil-Pelaez, Note on the inversion theorem, Biometrika, 38 (1951), 481–482.
[11]P. Kevei, Generalized n-Paul paradox, Statist. Probab. Lett., 77 (2007), 1043–1049. · Zbl 1138.60026 · doi:10.1016/j.spl.2006.08.027
[12]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
[13]Z. Megyesi, A probabilistic approach to semistable laws and their domains of partial attraction, Acta Sci. Math. (Szeged), 66 (2000), 403–434.
[14]G. Pap, The accuracy of merging approximation in generalized St. Petersburg games, preprint.
[15]V. V. Petrov, Limit Theorems of Probability Theory, Oxford Studies in Probability 4, Clarendon Press (Oxford, 1996).
[16]B. Rosén, On the asymptotic distribution of sums of independent identically distributed random variables, Ark. Mat., 4 (1962), 323–332. · Zbl 0103.11902 · doi:10.1007/BF02591508