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  • Autor
    • Berkes, István
  • TitelStrong approximation of the St. Petersburg game
  • Datei
  • DOI10.1080/02331888.2016.1269476
  • Erschienen inA Journal of Theoretical and Applied Statistics
  • Band51
  • Erscheinungsjahr2017
  • Heft1
  • Seiten3-10
  • LicenceCC BY
  • ISSN1029-4910
  • ZugriffsrechteCC-BY
  • Download Statistik534
  • Peer ReviewJa
  • AbstractLet \(X, X_1 , X_2 , . . .\) be i.i.d. random variables with \(P(X = 2^k) = 2^{-k} (k ∈ \mathbb{N}\) and let \(S_n = ∑^n_{k=1} X_k\) . The properties of the sequence \(S_n\) have received considerable attention in the literature in connection with the St. Petersburg paradox (Bernoulli 1738). Let \(\{Z(t), t ≥ 0\}\) be a semistable Lévy process with underlying Lévy measure \(∑_{k∈\mathbb{Z}} 2^{−k} δ_{2^k}\) . For a suitable version of \((X_k)\) and \(Z(t)\) , we prove the strong approximation \(S_n = Z(n) + O(n_{5/6+ε}) \)a.s. This provides the first example for a strong approximation theorem for partial sums of i.i.d. sequences not belonging to the domain of attraction of the normal or stable laws.