Paul L\'evy, strong approximation and the St. Petersburg paradox

Abstract

This paper discusses a forgotten remark of Paul L\'evy (1935), determining the asymptotic distribution of sums of i.i.d. random variables with tails $cx^{-\alpha}\psi(\log x)$, where $0<\alpha<2$ and $\psi$ is a periodic function on $\mathbb R$. Such sums occur in the St. Petersburg paradox and L\'evy's results precede the crucial results of Martin-L\"of (1985) and Cs\"org\H{o} and Dodunekova (1991) on the paradox by 50 years. L\'evy's proof uses a coupling argument similar to Skorohod representation and provides a strong (pointwise) approximation result, the first in probability theory. In the same paper, L\'evy also proves limit theorems for i.i.d. sums by using the quantile transform, another 'first' in the theory.