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Tail Probabilities of St. Petersburg Sums, Trimmed Sums, and Their Limit

Author

Listed:
  • István Berkes

    (Graz University of Technology)

  • László Györfi

    (Budapest University of Technology and Economics)

  • Péter Kevei

    (Technische Universität München)

Abstract

We provide exact asymptotics for the tail probabilities $${\mathbb {P}}\{ S_{n,r} > x \}$$ P { S n , r > x } as $$x \rightarrow \infty $$ x → ∞ , for fixed n, where $$S_{n,r}$$ S n , r is the r-trimmed partial sum of i.i.d. St. Petersburg random variables. In particular, we prove that although the St. Petersburg distribution is only O-subexponential, the subexponential property almost holds. We also determine the exact tail behavior of the r-trimmed limits.

Suggested Citation

  • István Berkes & László Györfi & Péter Kevei, 2017. "Tail Probabilities of St. Petersburg Sums, Trimmed Sums, and Their Limit," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1104-1129, September.
    • Handle: RePEc:spr:jotpro:v:30:y:2017:i:3:d:10.1007_s10959-016-0677-5
    DOI: 10.1007/s10959-016-0677-5
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    References listed on IDEAS

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    1. Csörgo, Sándor & Simons, Gordon, 1996. "A strong law of large numbers for trimmed sums, with applications to generalized St. Petersburg games," Statistics & Probability Letters, Elsevier, vol. 26(1), pages 65-73, January.
    2. Allan Gut & Anders Martin-Löf, 2016. "A Maxtrimmed St. Petersburg Game," Journal of Theoretical Probability, Springer, vol. 29(1), pages 277-291, March.
    3. Berkes, István & Csáki, Endre & Csörgo, Sándor, 1999. "Almost sure limit theorems for the St. Petersburg game," Statistics & Probability Letters, Elsevier, vol. 45(1), pages 23-30, October.
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    Cited by:

    1. Péter Kevei & Dalia Terhesiu, 2020. "Darling–Kac Theorem for Renewal Shifts in the Absence of Regular Variation," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2027-2060, December.

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