Paul Lévy, strong approximation and the St. Petersburg paradox

This paper discusses a forgotten remark of Paul Lévy (1935), determining the asymptotic distribution of sums of i.i.d. random variables with tails $cx^{-α}ψ(\log x)$, where $0<α<2$ and $ψ$ is a periodic function on $\mathbb R$. Such sums occur in the St. Petersburg paradox and Lévy's results precede the crucial results of Martin-Löf (1985) and Csörgő and Dodunekova (1991) on the paradox by 50 years. Lévy'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, Lévy also proves limit theorems for i.i.d. sums by using the quantile transform, another 'first' in the theory.

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