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Probability Proofs for Stirling (and More): the Ubiquitous Role of $\mathbf{\sqrt{2π}}$

Nils Lid Hjort, Emil Aas Stoltenberg

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. 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 $π$.

Probability Proofs for Stirling (and More): the Ubiquitous Role of $\mathbf{\sqrt{2π}}$

Abstract

The Stirling approximation formula for 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. 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 .

Paper Structure

This paper contains 12 sections, 1 theorem, 51 equations.

Table of Contents

  1. Introduction
  2. The CLT connection
  3. Poisson.
  4. Gammas.
  5. Binomial.
  6. Sums of uniforms and the Irwin--Hall distribution
  7. Stirling and Wallis from the median of a uniform sample
  8. Stirling, again.
  9. The Wallis product.
  10. Stirling via Laplace approximations
  11. Notes on the historical background for different themes
  12. Concluding remarks

Key Result

Lemma 1

For the Irwin--Hall density $f_n$, with $n$ even, we have

Theorems & Definitions (2)

  • Lemma 1
  • proof