A strong invariance principle for associated random fields (Q1775452)

Let \((X_j)_{j\in\mathbb Z^d_+}\) be a weakly stationary associated random field with zero mean and \(\rho(j-k)=EX_jX_k\), \(u(n)=\sum_{i\in\mathbb Z^d:\| i\|\geq n}\rho(i)\), \(u(0)<\infty\). Under the conditions \(\sup_{j\in\mathbb Z^d_+}E|X_j|^{2+r+\delta}<\infty\) for some \(r,\delta>0\) and \(u(n)= O(n^{-\nu})\), \(\nu>0\), the strong invariance principle is proved.