Stein's method for negatively associated random variables with applications to second order stationary random fields

Let $\mathit{\xi }=({\xi }_1,\dots ,{\xi }_m)$ be a negatively associated mean zero random vector with components that obey the bound $|\xi_i| \le B, i=1,\ldots,m$,andwhosesum$W = \sum_{i=1}^m \xi_i$ has variance 1, the bound $$d_1(\mathcal{L}(W),\mathcal{L}(Z))\le 5B-5.2\sum _{i≠j}{\sigma }_{ij}\mathrm{.}$$ is obtained where $Z$ has the standard normal distribution and $d_1(\cdot ,\cdot )$ is the $L^1$ metric. The result is extended to the multidimensional case with the $L^1$ metric replaced by a smooth functions metric. Applications to second order stationary random fields with exponential decreasing covariance are also presented.