  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.
