Strong approximation and a central limit theorem for St. Petersburg sums

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The St. Petersburg paradox (Bernoulli, 1738) concerns the fair entry fee in a game where the winnings are distributed as P(X=2k)=2k,k=1,2,…. The tails of X are not regularly varying and the sequence Sn of accumulated gains has, suitably centered and normalized, a class of semistable laws as subsequential limit distributions (Martin-Löf, 1985; Csörgő and Dodunekova, 1991). This has led to a clarification of the paradox and an interesting and unusual asymptotic theory in past decades. In this paper we prove that Sn can be approximated by a semistable Lévy process {L(n),n≥1} with a.s. error O(√n(logn)1+ε) and, surprisingly, the error term is asymptotically normal, exhibiting an unexpected central limit theorem in St. Petersburg theory.

Original languageEnglish
Pages (from-to)4500-4509
Number of pages10
JournalStochastic Processes and their Applications
Issue number11
Publication statusPublished - Nov 2019


  • Central limit theorem
  • Semistable process
  • St. Petersburg sums
  • Strong approximation

ASJC Scopus subject areas

  • Statistics and Probability
  • Modelling and Simulation
  • Applied Mathematics


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