• Autor
    • Berkes, István
  • TitelStrong approximation and a central limit theorem for St. Petersburg sums
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  • LicenceCC BY
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  • AbstractThe St. Petersburg paradox (Bernoulli 1738) concerns the fair entry fee in a game where the winnings are distributed as  \(P(X = 2^k) = 2^{−k} , k = 1, 2, . . .\). The tails of X are not regularly varying and the sequence \(S_n\) of accumulated gains has a remarkable asymptotic behavior: as Martin-Löf (1985) and Csörgő and Dodunekova (1991) showed, \(S_n /n − log_2 n\) has a class of semistable laws as subsequential limit distributions. 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 \(S_n\) can be approximated by a semistable Lévy process \(\{L(n), n ≥ 1\}\) with a.s. error \(O(\sqrt{n}(log n)^{1+ε})\) and, surprisingly, the error term is asymptotically normal, exhibiting an unexpected central limit theorem in St. Petersburg theory.