Probability and moment inequalities for weakly dependent double-indexed random variables (Q2784987)

Let \(|V_1-V_2|\) be the Euclidean distance between subsets in \(R^{N}\) and let \(\xi(\vec i)\) be random variables indexed by \(\vec i\in Z_{+}^{N}\). Denote the sigma-field \(\sigma\{\xi(\vec i), \vec i\in V\}\) by \(\sigma(V)\). The uniformly strong mixing coefficient is defined as NEWLINE\[NEWLINE\varphi_{\xi}^{N}(r)= \sup\limits_{A\in\sigma(V_1),B\in\sigma(V_2),|V_1-V_2|\geq r}\left\{\left|{P(AB)\over P(A)}-P(B) \right|, P(A)\neq 0\right\}.NEWLINE\]NEWLINE The author gives an example of a random field that has uniform strong mixing coefficients decreasing to zero while this field is not a collection of independent random variables. The probability and the moment inequalities are obtained for double-indexed weakly dependent random variables and for their sums.