On normal approximation for strongly mixing random fields

In this paper, we estimate the difference $|\mathbf Eh(Z_V)-\mathbf Eh(N)|$, where $Z_V$ is a sum over any finite subset $V$ of the standard lattice $\mathbf Z^d$ of normalized random variables of the strongly mixing random field $\{X_a,\ a\in\mathbf Z^d\}$ (without assuming stationarity) and $N$ is a standard normal random variable for the function $h\colon\mathbf R\to\mathbf R$, which is finite and satisfies the Lipschitz condition. In a particular case, the obtained upper bounds of $|\mathbf Eh(Z_V)-\mathbf Eh(N)|$ in Theorems 3 and 4 are of order $O(|V|^{-1/2})$.