Strong approximation of the St. Petersburg game

István Berkes*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Let X, X1, X2,... be i.i.d. random variables with P(X = 2k)=2-k (k∈ℕ) and let Sn=∑nk=1Xk. The properties of the sequence Sn 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 ∑kεℤ2-kδ2k. For a suitable version of (Xk) and Z(t), we prove the strong approximation Sn=Z(n)+O(n5/6+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.
Original languageEnglish
Pages (from-to)3-10
Number of pages8
Issue number1
Publication statusPublished - 3 Jan 2017


  • a.s. invariance principle
  • semistable laws
  • St. Petersburg game

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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