PreprintCC0arXiv
A clarification of Fermi's formula for the maximum of partial sums
Abstract
In Fermi's 1926 note, the expression assigned to the probability that all partial sums remain below a threshold is the complementary crossing probability. We obtain the correct probability from the same diffusion equation and locate the inconsistency in the final integration. We then state the limit for independent increments under the Lindeberg condition. A counterexample with smooth densities shows that small individual variances do not suffice to justify this limit. The probabilistic results are classical; the purpose of this note is to reconcile the stated event with the printed calculation and to make the sufficient hypotheses for the discrete limit explicit.
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BibTeX
@misc{lodi2026clarification,
title = {{A clarification of Fermi's formula for the maximum of partial sums}},
author = {Alessandro Lodi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15935},
eprint = {2610.15935},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15935},
note = {CC0},
}
RIS
TY - UNPB
TI - A clarification of Fermi's formula for the maximum of partial sums
AU - Alessandro Lodi
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15935
UR - https://arxiv.org/abs/2610.15935
AB - In Fermi's 1926 note, the expression assigned to the probability that all partial sums remain below a threshold is the complementary crossing probability. We obtain the correct probability from the same diffusion equation and locate the inconsistency in the final integration. We then state the limit for independent increments under the Lindeberg condition. A counterexample with smooth densities shows that small individual variances do not suffice to justify this limit. The probabilistic results are classical; the purpose of this note is to reconcile the stated event with the printed calculation and to make the sufficient hypotheses for the discrete limit explicit.
ER -