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

TL;DR

This paper develops Stein's method for negatively associated random variables, providing bounds on normal approximation errors, extending to multidimensional cases, and applying to stationary random fields with decaying covariance.

Contribution

It introduces new bounds for normal approximation of negatively associated variables and extends Stein's method to multidimensional and stationary random fields.

Findings

Derived explicit bounds for the normal approximation error in the $L^1$ metric.

Extended results to multidimensional settings with smooth functions metrics.

Applied the method to second order stationary random fields with exponential decay in covariance.

Abstract

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.