Some examples of mixing random fields
Let \(X=(x_ t,t \in \mathbb{Z}^ d)\) be a strictly stationary random field on probability space \((\Omega, {\mathcal F}, P)\), \(\alpha ({\mathcal A}, {\mathcal B})\) be a Rosenblatt's measure of dependence of two \(\sigma\)-fields \({\mathcal A}\) and \({\mathcal B}\), \[ \rho ({\mathcal A}, {\mathcal B})=\sup \bigl\{ | \text{corr} (f,g) |,\;f \in L_ 2 ({\mathcal A}),\;g \in L_ 2({\mathcal B}) \bigr\}, \] \[ \beta({\mathcal A}, {\mathcal B})=\sup {1 \over 2} \sum^ I_{i=1} \sum^ I_{j=1} \bigl | P(A_ iB_ j)-P(A_ i)P(B_ j) \bigr |, \] where the last sup is taken over all pairs of partitions \(\{A_ 1,\dots,A_ I\}\) and \(\{B_ 1,\dots,B_ I\}\) of \(\Omega\) such that \(A_ i \in {\mathcal A}\), \(B_ j \in {\mathcal B}\). Let \[ \begin{aligned} \alpha_{j,k} (X,r) & = \sup \alpha \bigl( \sigma (X_ t,\;t \in S),\;\sigma (X_ t,\;t \in T) \bigr), \\ \rho_{j,k} (X,r) & = \sup \rho \bigl( \sigma (X_ t,\;t \in S),\;\sigma (X_ t,\;t \in T) \bigr), \\ \beta_{j,k} (X,r) &= \sup \beta \bigl( \sigma (X_ t,\;t \in S),\;\sigma(X_ t,\;t \in T) \bigr), \end{aligned} \] where all three sups are taken over all pairs of disjoint nonempty subsets \(S,T \subseteq \mathbb{Z}^ d\) such that \(\text{dist} (S,T) \geq r\), \(\text{card} S \leq j\), \(\text{card} T \leq k\). Suppose that \(L_ 1,L_ 2,\dots\) is a sequence of nonzero elements of \(\mathbb{Z}^ d\) such that for all \(n \geq 1\), \(\| L_ n \| \leq \| L_{n+1} \|\). Define a sequence of positive numbers \(r_ 1,r_ 2,\dots\) by \(r_ n=\| L_ n \|\). Suppose that \(c_ 1,c_ 2,c_ 3,\dots\) is a sequence of positive numbers, such that \(c_ 1 \leq 1/s\) and \(c_{n+1} \leq c_ n/2\) for all \(n \geq 1\), \(0<q \leq 1\). Then there exists a random field \(X\) with the following properties: (1) The distribution of \(X_ 0\) does not have any atoms. (2) For all \(n \geq 1\), \(\alpha( \sigma (X_ 0),\sigma (X_{L(n)})) \geq qc_ n/2\) and \(\rho (\sigma (X_ 0), \sigma(X_{L(n)}))=q\). (3) For each \(n \geq 1\), each integer \(k\) such that \(1 \leq k \leq {1 \over s} c_ n\) one has \(\alpha_{k,k}\) \((X,r_ n)/k \geq qc_ n/2\). (4) For each \(n \geq 1\), each \(k\geq 1\) one has \(\beta_{k,\infty} (X,r_ n)/k \leq 8qc_ n\). (5) For all \(r \geq 1\), \(\alpha_{\infty,\infty}\) \((X,r) \geq q/(2\pi)\). (6) For all \(r \geq 1\), \(\rho_{1,1}(X,r)=\rho_{\infty,\infty} (X,r)=q\).
scientific article; zbMATH DE number 515186
scientific article