A Moment Generating Function Proof of the Lindeberg-Levy Central Limit Theorem

The central limit theorem (CLT) commonly presented in introductory probability and mathematical statistics courses is a simplification of the Lindeberg-Levy 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-Levy 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.