Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The limiting distribution of the St. Petersburg game


Author: Ilan Vardi
Journal: Proc. Amer. Math. Soc. 123 (1995), 2875-2882
MSC: Primary 60F05
DOI: https://doi.org/10.1090/S0002-9939-1995-1322939-3
MathSciNet review: 1322939
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The St. Petersburg game is a well-known example of a random variable which has infinite expectation. Csörgő and Dodunekova have recently shown that the accumulated winnings do not have a limiting distribution, but that if measurements are taken at a subsequence $ {b_n}$, then a limiting distribution exists exactly when the fractional parts of $ {\log _2}{b_n}$ approach a limit. In this paper the characteristic functions of these distributions are computed explicitly and found to be continuous, self-similar, nowhere differentiable functions.


References [Enhancements On Off] (What's this?)

  • [1] Sándor Csörgő and Rossitza Dodunekova, Limit theorems for the Petersburg game, Sums, trimmed sums and extremes, Progr. Probab., vol. 23, Birkhäuser Boston, Boston, MA, 1991, pp. 285–315. MR 1117274
  • [2] Hubert Delange, Sur la fonction sommatoire de la fonction“somme des chiffres”, Enseignement Math. (2) 21 (1975), no. 1, 31–47 (French). MR 0379414
  • [3] William Feller, An introduction to probability theory and its applications. Vol. I, John Wiley and Sons, Inc., New York; Chapman and Hall, Ltd., London, 1957. 2nd ed. MR 0088081
  • [4] P. Flajolet, P. Grabner, and P. Kirschenhofer, Mellin transforms and asymptotics: Digital sums, Theoretical Computer Science (in press).
  • [5] Lothar Heinrich, Rates of convergence in stable limit theorems for sums of exponentially 𝜓-mixing random variables with an application to metric theory of continued fractions, Math. Nachr. 131 (1987), 149–165. MR 908807, https://doi.org/10.1002/mana.19871310114
  • [6] I. A. Ibragimov and Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov; Translation from the Russian edited by J. F. C. Kingman. MR 0322926
  • [7] A. Khintchine, Metrische Kettenbruchprobleme, Compositio Math. 1 (1935), 361–382 (German). MR 1556899
  • [8] Anders Martin-Löf, A limit theorem which clarifies the “Petersburg paradox”, J. Appl. Probab. 22 (1985), no. 3, 634–643. MR 799286
  • [9] Walter Philipp, Limit theorems for sums of partial quotients of continued fractions, Monatsh. Math. 105 (1988), no. 3, 195–206. MR 939942, https://doi.org/10.1007/BF01636928

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 60F05

Retrieve articles in all journals with MSC: 60F05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1322939-3
Article copyright: © Copyright 1995 American Mathematical Society