The accuracy of the Gaussian approximation to the sum of independent variates

The sum of finitely many variates possesses, under familiar conditions, an almost Gaussian probability distribution. This already much discussed central limit theorem (x) in the theory of probability is the object of further investigation in the present paper. The cases of Liapounoff (2), Lindeberg (3), and Feller (4) will be reviewed. Numerical estimates for the degrees of approximation attained in these cases will be presented in the three theorems of §4. Theorem 3, the arithmetical refinement of the general theorem of Feller, constitutes our principal result. As the foregoing implies, we require throughout the paper that the given variates be totally independent. And we consider only one-dimensional variates. The first three sections of the paper are devoted to the preparatory Theorem 1 in which the variates. meet the further condition of possessing finite third order absolute moments. Let X\, Xi, • • •, Xn be the given variates. For each k{k = \, 2, ■ ■ ■, n) let ^ (Xk) and ixs{Xk) denote, respectively, the second and third order absolute moments of Xk about its mean (expected) value a*. These moments are either both zero or both positive. The former case arises only when Xk is essentially constant, i. e. , differs from its mean value at most in cases of total probability zero. To avoid trivialities we suppose that PziXk) >0 for at least one k (k = 1, 2, • • •, n). The non-negative square root of m{Xk) is the standard deviation of Xk and will be denoted by ak. We call