In a recent paper, C. Stein has given a new, direct technique for bounding the error of the normal approximation to the distribution of a sum of dependent random variables, assuming the variables form a stationary sequence with eighth moments. In the present paper we give two $L_1$ bounds on this error for an arbitrary $m$-dependent sequence with second moments.
"$L_1$ Bounds for Asymptotic Normality of $m$-Dependent Sums Using Stein's Technique." Ann. Probab. 2 (3) 522 - 529, June, 1974. https://doi.org/10.1214/aop/1176996670