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Are there any interesting examples of second order stationary processes on ${\mathcal R}^2$ or ${\mathcal R}^3$ that are not isotropic? The book I am looking at has no examples.

Update : I asked this question in mathoverflow, apparently such examples are not easy to come by.

Update: Processes with anisotropic variograms are examples of non-isotropic stationary processes.

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Well yes, lots of them. Consider for example some independent stationary processes $U=(U_x)_{x\in\mathbb R}$ and $V=(V_y)_{y\in\mathbb R}$ and define $X_{x,y}=U_x+V_y$ for every $(x,y)$ in $\mathbb R^2$. Then $X=(X_{x,y})_{(x,y)\in\mathbb R^2}$ is invariant under the effect of every translation but not under the effect of the rotation of angle $\frac\pi2$, that is, unless $U$ and $V$ are identically distributed.

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