Probability Proofs for Stirling (and More): The Ubiquitous Role of 2π

Nils Lid Hjort and Emil Aas Stoltenberg

The American Statistician, 2026, vol. 80, issue 3, 405-412

Abstract: The Stirling approximation formula for n! dates from 1730. Here we give new and instructive proofs of this and related approximation formulae via tools of probability and statistics. There are connections to the Central Limit Theorem and also to approximations of marginal distributions in Bayesian setups, with arguments which can be worked through by Master and PhD level students (and above). Certain formulae emerge by working through particular instances, some independently verifiable but others perhaps not. A particular case yielding new formulae is that of summing independent uniforms, related to the Irwin–Hall distribution. Yet further proofs of the Stirling flow from examining aspects of limiting normality of the sample median of uniforms, and from these again we find a proof for the Wallis product formula for π. A section detailing historical aspects and development is included, from Wallis 1656 and de Moivre and Stirling 1730 to Laplace 1778, etc.

Date: 2026
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DOI: 10.1080/00031305.2025.2603256

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