On the distribution function for a function ofnstatistic variables and the Central Limit Theorem in the Mathematical Theory of Probability

Abstract

According to the Central Limit Theorem in the mathematical theory of probability, the sum i. e. a specific function of n mutually independent statistic variables — under very general conditions — converges towards the normal distribution function : when the number of the statistic variables entering into the sum (function) examined tends to infinity. This theorem has attracted the attention of many mathematically interested statisticians. A great number of famous mathematicians, as Moivre, Laplace, Ljapounoff, Lindeberg, Kolmogoroff, Khintchine, Petrowsky, Lévy, Feller, Cramér etc. have dealt with the formulation of and the proof of the central limit theorem. As a result of these efforts, the space of validity of the central limit theorem has been found to be very extensive and very difficult to define, so that the limits of the space within which the theorem provably is valid have repeatedly been widened. Feller and Lévy have, however, found a formulation touching the limits for the space of validity of the theorem in the case that only considers the sum of n mutually independent variables, the relative signification of which approaches zero.