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

The 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, 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 S_n 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.