Revisiting De Moivre-Laplace

TL;DR

This paper presents an accessible proof of the de Moivre-Laplace theorem suitable for undergraduates, avoiding complex series and Stirling's approximation, and introduces a non-asymptotic inequality linking binomial and Gaussian distributions.

Contribution

It offers a simplified, rigorous proof of the de Moivre-Laplace theorem using basic inequalities and exact formulas, making the result more approachable for educational purposes.

Findings

Provides an undergraduate-friendly proof of the de Moivre-Laplace theorem.

Derives a non-asymptotic inequality linking binomial and Gaussian distributions.

Avoids complex series expansions and Stirling's formula in the proof.

Abstract

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 $\exp(t)\geq 1+t$. 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.