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

Abstract 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-Lof, 1985; Csorgő 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 Levy process { L ( n ) , n ≥ 1 } with a.s. error O ( n ( log n ) 1 + e ) and, surprisingly, the error term is asymptotically normal, exhibiting an unexpected central limit theorem in St. Petersburg theory.