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

Abstract

Let $\mathbf{\xi }=({\xi }_1,\dots ,{\xi }_m)$ be a negatively associated mean zero random vector with components that obey the bound $|{\xi }_i|\le B,i=1,\dots ,m$, and whose sum $W={\sum }_{i=1}^m{\xi }_i$ has variance 1, the bound \[ d_1\big({\cal L}(W),{\cal L}(Z)\big) \le 5B - 5.2\sum_{i \not = j} σ_{ij}. \] 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.

19 pages. arXiv admin note: text overlap with arXiv:1603.05322

References in corpus (1)

• Stein's method for positively associated random variables with applications to the Ising and voter models, bond percolation, and contact process

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