ON NORMAL APPROXIMATION RATES FOR CERTAIN SUMS OF DEPENDENT RANDOM-VARIABLES

Let X(1),...,X(n) be dependent random variables, and set lambda = E {Sigma(i=1)(n) X(i)}, and sigma(2) = Var{Sigma(i=1)(n) X(i)}. In most of the applications of Stein's method for normal approximations, the error rate \P((Sigma(i=1)(n) X(i) - lambda)/sigma less than or equal to w)- Phi(w)\ is of the order of sigma(-1/2). This rate was improved by Stein (1986) and others in some special cases. In this paper it is shown that for certain bounded random variables, a simple refinement of error-term calculations in Stein's method leads to improved rates.