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Limit Theorems for the Petersburg Game

  • Sándor Csörgő
  • Rossitza Dodunekova
Part of the Progress in Probability book series (PRPR, volume 23)

Abstract

We determine all possible subsequences \(\left\{ {n_k } \right\}_{k = 1}^\infty\)of the positive integers for which the suitably centered and normalized total gain S nk in n k Petersburg games has an asymptotic distribution as k → ∞, and identify the corresponding set of Hmiting distributions. We also solve all the companion problems for lightly, moderately, and heavily trimmed versions of the sum S nk and for the respective sums of extreme values in S nk .

Keywords

Limit Theorem Asymptotic Distribution Total Gain Partial Attraction Divisible Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    S. CSÖRGŐ, An extreme-sum approximation to infinitely divisible laws without a normal component. In: Probability on Vector Spaces IV. (S. Cambanis and A. Weron, eds.) pp. 47–58. Lecture Notes in Mathematics 1391, Springer, Berlin, 1989.CrossRefGoogle Scholar
  2. [2]
    S. CSÖRGÖ, A probabilistic approach to domains of partial attraction. Adv. in Appl Math. 11 (1990), to appear.Google Scholar
  3. [3]
    S. CSÖRGÖ, E. HAEUSLER, and D. M. MASON, A probabilistic approach to the asymptotic distribution of sums of independent, identically distributed random variables. Adv. in Appl. Math. 9(1988), 259–333.MathSciNetCrossRefGoogle Scholar
  4. [4]
    S. CSÖRGÖ, E. HAEUSLER, and D. M. MASON, The asymptotic distribution of trimmed sums. Ann. Probab. 16(1988), 672–699.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    S. CSÖRGÖ, E. HAEUSLER, and D. M. MASON, The asymptotic distribution of extreme sums. Ann. Probab. 19 (1991), to appear.Google Scholar
  6. [6]
    S. CSÖRGÖ, E. HAEUSLER, and D. M. MASON, The quantile - transform - empirical - process approach to limit theorems for sums of order statistics. In: Sums, Trimmed Sums, and Extremes (M. G. Hahn, D. M. Mason, and D. C. Weiner, eds.). Birkhäuser, Basel, 1990.Google Scholar
  7. [7]
    S. CSÖRGÖ and D. M. MASON, Intermediate sums and stochastic compactness of maxima. Submitted.Google Scholar
  8. [8]
    W. FELLER, An Introduction to Probability Theory and its Applications, Vol. l. Wiley, New York, 1950.MATHGoogle Scholar
  9. [9]
    P. LEVY, Theorie de Vaddition des variables aleatoires, ed. Gauthier - Villars, Paris, 1954.Google Scholar
  10. [10]
    A. MARTIN - LÖF, A limit theorem which clarifies the ‘Petersburg paradox’. J. Appl Probab. 22(1985), 634–643.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    G. SHAFER, The St. Petersburg paradox. In:Encyclopedia of Statistical Sciences, Vol 8 (S. Kotz, N. L. Johnson, and C. B. Read, eds.), pp. 865–870. Wiley, New-York, 1988.Google Scholar

Copyright information

© Birkhäuser Boston 1991

Authors and Affiliations

  • Sándor Csörgő
    • 1
  • Rossitza Dodunekova
    • 2
  1. 1.Department of StatisticsThe University of MichiganAnn ArborUSA
  2. 2.Department of Probability and StatisticsUniversity of SofiaSofiaBulgaria

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