The Central Limit Theorem, one of the most striking and useful results in probability and statistics, explains why the normal distribution appears in areas as diverse as gambling, measurement error, sampling, and statistical mechanics. In essence, the Central Limit Theorem states that the normal distribution applies whenever one is approximating probabilities for a quantity which is a sum of many independent contributions all of which are roughly the same size. It is the Lindeberg-Feller Theorem which makes this statement precise in providing the sufficient, and in some sense necessary, Lindeberg condition whose satisfaction accounts for the ubiquitous appearance of the bell shaped normal. Generally the Lindeberg condition is handled using Fourier methods and is somewhat hard to interpret from the classical point of view. Here we provide a simpler, equivalent, and more easily interpretable probabilistic formulation of the Lindeberg condition and demonstrate its sufficiency and partial necessity in the Central Limit Theorem using more elementary means. The seeds of the Central Limit Theorem, or CLT, lie in the work of Abraham de Moivre, who, in 1733, not being able to secure himself an academic appointment, supported himself consulting on problems of probability, and gambling. He approximated the limiting probabilities of the Binomial distribution, the one which governs the behavior of the number Sn of success in an experiment which consists of n independent trials, each one having the same probability p ∈ (0, 1) of success. Each individual trial of the experiment can be modelled byX, a (Bernoulli) random variable which records one for each success, and zero for each failure,
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