On normal approximation rates for certain sums of dependent random variables

Abstract

Let X 1, …, X n be dependent random variables, and set λ = E∑ n i=1 X i , and σ 2 = Var∑ n i=1 X i . In most of the applications of Stein's method for normal approximations, the error rate | P((∑ n i=1 X i − λ)/σ ⩽ w) − Φ( w)| is of the order of σ − 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.