Lévy and Feller on Normal Limit Distributions around 1935
Lévy [1924, 17] had already stated that Lindeberg‘s condition for the CLT accorded particularly well with “la nature des choses.” Was he trying to say that, in a certain way, this conditionwas also necessary for convergence to the normal distribution? In fact, more than 10 years would pass before Lévy and Feller almost simultaneously proved that certain conditions are both sufficient and necessary for the convergence of distributions of suitably normed sums of independent random variables to the normal distribution. These examinations foresaw general normings that no longer assumed the existence of the variance, and it emerged within this framework that, i the classical case, the Lindeberg condition is necessary for the CLT if the influence of the individual random variables on their sum can be asymptotically neglected in a particular sense. The possibility of nonclassical norming, which Bernshtein and Lévy had already addressed in the 1920s, out of a desire to exhaust all analytical possibilities, became all the more interesting the further mathematical probability theory departed from its original areas of application.
KeywordsIndependent Random Variable Standard Normal Distribution Elementary Error Distribution Versus Decomposition Principle
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