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

Let $\boldsymbol{\xi}=(\xi_1,\ldots,\xi_m)$ be a negatively associated mean zero random vector with components that obey the bound $|\xi_i| \le B, i=1,\ldots,m$, and whose sum $W = \sum_{i=1}^m \xi_i$ has variance 1, the bound $$d_1\big({\cal L}(W),{\cal L}(Z)\big) \le 5B - 5.2\sum_{i \not = j} \sigma_{ij}.$$ 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.