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Revisiting De Moivre-Laplace

Raphaël Cerf

TL;DR

The paper develops an elementary, undergraduate-friendly proof of the $de Moivre-Laplace$ theorem for the binomial distribution, avoiding uniform convergence arguments and Stirling’s formula. It replaces series and log-expansions with the simple inequality $\exp(t)\geq 1+t$ and leverages Wallis integrals to connect binomial probabilities to the normal law, producing both upper and lower bounds and non-asymptotic inequalities. The symmetric case ($p=1/2$) is analyzed in depth, with explicit bounds on the central binomial probabilities and a clean limit to the standard normal CDF, while the odd-case and the general $p\in(0,1)$ extension are described using analogous decompositions and bounds. The results yield a pedagogically transparent pathway to the central limit phenomenon for the binomial, including quantitative non-asymptotic controls that quantify the binomial-to-Gaussian proximity without Stirling's formula, enhancing undergraduate probability curricula and intuition around distributional convergence.

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.

Revisiting De Moivre-Laplace

TL;DR

The paper develops an elementary, undergraduate-friendly proof of the theorem for the binomial distribution, avoiding uniform convergence arguments and Stirling’s formula. It replaces series and log-expansions with the simple inequality and leverages Wallis integrals to connect binomial probabilities to the normal law, producing both upper and lower bounds and non-asymptotic inequalities. The symmetric case () is analyzed in depth, with explicit bounds on the central binomial probabilities and a clean limit to the standard normal CDF, while the odd-case and the general extension are described using analogous decompositions and bounds. The results yield a pedagogically transparent pathway to the central limit phenomenon for the binomial, including quantitative non-asymptotic controls that quantify the binomial-to-Gaussian proximity without Stirling's formula, enhancing undergraduate probability curricula and intuition around distributional convergence.

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 . 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.
Paper Structure (11 sections, 1 theorem, 115 equations, 2 figures)

This paper contains 11 sections, 1 theorem, 115 equations, 2 figures.

Table of Contents

  1. Introduction.
  2. de Moivre theorem.
  3. Upper bound.
  4. Lower bound.
  5. Conclusion.
  6. The case of odd integers.
  7. Avoiding Stirling's formula.
  8. Non-asymptotic inequalities.
  9. de Moivre--Laplace theorem.
  10. Upper bound.
  11. Lower bound.

Key Result

Theorem 1

Let $p\in]0,1[$. For each $n\geq 1$, let $X_1,\dots,X_n$ be $n$ independent Bernoulli random variables with parameter $p$, i.e., For any real numbers $a<b$, the following limit holds:

Figures (2)

  • Figure 1: The laws $B(n,\frac{1}{2})$, ${\mathcal{N}}(\frac{n}{2},\frac{n}{4})$ for $n=1,2,3,4,5,6,7,8,20,40,50,100$
  • Figure 2: The Gaussian distribution ${\mathcal{N}}(0,1)$

Theorems & Definitions (1)

  • Theorem 1: de Moivre--Laplace