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

TL;DR

This paper revisits Paul Lévy's 1935 work on the asymptotic behavior of sums of i.i.d. variables with heavy tails, highlighting its pioneering role in strong approximation and limit theorems related to the St. Petersburg paradox.

Contribution

It uncovers Lévy's early results on heavy-tailed sums, coupling methods, and limit theorems, predating and informing later developments in probability theory.

Findings

Lévy's asymptotic distribution results for heavy-tailed sums

First strong (pointwise) approximation in probability theory

Limit theorems using the quantile transform for i.i.d. sums

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.