JOURNAL OF APPROXIMATION THEORY 13, 327-340 (1975) On the Rate of Approximation in the Central Limit Theorem P. L. BUTZER, L. HAFIN,* AND U. WESTPHAL DEDICATED TO PROFESSOR G. 0. LORENTZ ON THE OCCASIOK OF HIS SIXTY-FIFTH BIRTHDAY 1. INTRODUCTION AND HISTORY Let (Q, ?I, Pr) be an arbitrary probability space with distribution function (d.f.) F,- of the real random variable (T.v.) X: 5-? --?’ R, delined by F,(x) -:= Pr{w E fin: X(w) t< xl, for every x E R. Let X* be a normally distributed random variable with mean 0 and variance 1, i.e., X* is a random variable with d.f. FX*(x) (27r-1/2 J-TC, exp(-U2/2) &. A sequence (X&, of real r.v.‘s with variance satisfying 0 < Var(X,) < +m, for each 11 E Iv, is said to satisfy the central limit theorem [2, p. 2231 in case (n ---f @I where FT,,(X) -+ Fx*(-4 (for each s E R), (1.1) (FTn(x) denoting the d.f. of the normalized sum 7,) and the expectation E(X) : = JR x SX(x)). This theorem is actually satisfied provided the sequence of r.v. is independent (which is case below) and identica!ly distributed. Of the many versions equivalent to (1.1) let us recall two further ones needed below. One is in terms of the pointwise convergence of the corre- sponding characteristic functions, namely * The research of Lothar Hahn was partially supported by UF‘G grant Ne 171’1. 327