2025/12/26 by Raphaël Cerf, Cerf, Raphaël
Engineering · Mathematics · #FOS: Mathematics #History and Theory of Mathematics #Probability (math.PR) #Probability and Statistical Research #Sports Dynamics and Biomechanics
paper · doi:10.48550/arxiv.2512.22330
openalex publication_date 2025/12/26 · openalex created_date 2025/12/31 · openalex updated_date 2026/07/28
We revisit the proof of the de Moivre--Laplace theorem, which is the ancestor of the central limit theorem for the binomial distribution. Our goal is to provide a proof that can be reasonably presented to undergraduate students within a basic course of probability theory. We follow the strategies presented in two classical references, the books of Breiman and Feller, but we replace the arguments involving series expansions of the logarithm or the exponential by the basic inequality exp(t)≥ 1+t. This way we avoid completely the use of uniform convergence and power series. We also avoid using Stirling's formula, instead we use the exact formula for the Wallis integral. As a by product of the proof, we also obtain a non-asymptotic inequality linking the binomial and the Gaussian distributions.