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

Nils Lid Hjort, Emil Aas Stoltenberg

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 statist…