A Moment Generating Function Proof of the Lindeberg–Lévy Central Limit Theorem

Abstract

The central limit theorem (CLT) commonly presented in introductory probability and mathematical statistics courses is a simplification of the Lindeberg–Lévy CLT which uses moment generating functions (mgf’s) in place of characteristic functions. As a result, it requires the existence of the mgf and, therefore, all moments. This article provides a new moment generating function proof of Lindeberg–Lévy which does not weaken it by requiring the existence of the mgf or higher order moments of the constituent random variables. The proof, which is accessible to first-year graduate students, provides an interesting application of Slutsky’s Theorem.