ABSTRACT
Let XI. . . . , X , be independent indicator random variables with P ( X , = 1) = 1 - P ( X , = 0) = p, , i = 1, .... n. Define W = Er=, X, , W( ' ) = W - X,, X = E,"=, p , and 2 to be a Poisson random variable with mean A. Let f h be the solution (which is unique except a t 0) of the Stein equation
Xf(w + 1) - wf(w) = h(w) - E h ( 2 ) where h is a bounded real-valued function defined on Z' = {0,1,2, . . .}. Then we have
where AJ(w) f (711 + l ) - f (711). A result of fhb011r ant1 Eaglcsori (1983) states that jiA fl,li, 5 2(1 A X ')/jl~/i,. Applying this reslrlt,, we obtain
wkierc dTv denotes the total variation distance. It is known that, the absolutc constant l is hest possil-)le arid the factor (1 A X-' ) has the correct orctcr for both s~rlall and large values of X. The sigriificancc of the factor (1 A X-') is explainctl in Chapter l of Barbour, I-iolst, a.nd ,Tanson (1 992).
