Asymptotic normality of random fields of positively or negatively associated processes (Q1333196)

The r.v.s \(Y_ 1,\dots,Y_ m\) are said to be positively associated (PA) if for every \(F,G:R^ m \to R\) increasing and such that \(EF^ 2 (Y_ 1, \dots, Y_ m) < \infty\), \(EG^ 2(Y_ 1, \dots, Y_ m) < \infty\), it holds that \(\text{cov} (F(Y_ 1, \dots, Y_ m)\), \(G(Y_ 1, \dots, Y_ m)) \geq 0\). The r.v.s \(Y_ 1, \dots, Y_ m\) are said to be negatively associated (NA) if for every nonempty subset \(A\) of \(\{1, \dots, m\}\) and for every \(F:R^ A \to R\), \(G:R^{A^ c} \to R\), which are increasing and such that \(EF^ 2 (Y_ i, i \in A) < \infty\), \(EG^ 2 (Y_ j,j \in A^ c)< \infty\) it holds that \(\text{cov} (F(Y_ i,i \in A)\), \(G(Y_ j,j \in A^ c) \geq 0\). Let \(X_ n\), \(n \in \mathbb{Z}^ d\), be a real-valued r.v. with finite second moment and subjet to covariance invariance and finite susceptibility. Under the assumption of PA or NA, asymptotic normality is established.