ON BOUNDS TO THE RATE OF CONVERGENCE IN THE CENTRAL LIMIT THEOREM

s | be independent and identically distributed random variables with zero mean, unit variance and finite third absolute moment y. The Berry-Esseen theorem states that sup p \ n'i £ Xj < x-<D(x) L j = i J Cyn-i and, because of the possibility of lattice valued variables Xj, no improvement in the rate on the right hand side can be obtained, even though the rate, for many Xj-distributions, is really much faster. In this paper, the departure of the distribution of a sum of independent but not necessarily identically distributed random variables from the normal is measured by a smoother metric than the uniform metric above, enabling a precise rate of convergence to be determined, within a tolerance which, in the above setting, is of order n~'. The rate of convergence in the central limit theorem is traditionally measured in terms of the uniform metric. For example, the Berry-Esseen theorem [4, p. I l l] states that A,, = sup [ I, ' • *']-where Yl,...,Yn are independent variables with zero means and satisfying n Y, E{Yf) = 1, and where <b is the standard normal distribution function. As is well-I known, inequalities such as this cannot be substantially improved unless smoothness conditions are imposed on the summands [4, p. 112]. However, we shall show that this awkwardness disappears if we use a "smooth " description of the rate of convergence. f " 1 Let Fa(x) = P \ X Yi**x \ and set d2{Fn, <D) = sup | f hdFn- J hd<b\/\\h\\2, where L=l J l>eC2 \\h\\2 = 11̂11 + P'll + P'll a nd IÎH = sup \h(x)\. Convergence in the metric d2 is X equivalent to weak convergence. Smooth metrics such as d2 would seem more natural in a theoretical setting than the unsmooth uniform metric, whose principal importance is in applications to the traditional theory of significance testing. And even here, the two-point action space (accept or reject the null hypothesis), which makes estimates of P £ Yt ^ x \ important in the construction of tests, leads not only to mathematical awkwardness, but also to an analogous philosophica