A Frequency-function form of the central limit theorem

Abstract

The central limit theorem in the calculus of probability has been extensively studied in recent years. In its simplest form the theorem states that if X1, X2,… is a sequence of independent, identically distributed random variables of mean zero, then under general conditions the distribution function of Zm = (X1 + … + Xn)/√ n converges as n → ∞ to the normal or Gaussian distribution function. This form of the theorem in terms of distribution functions is the one required in statistical work, since it enables statements to be made about the limiting behaviour of prob {a ≤ Zn ≤ b}.