Scaling limits of linear random fields on ${\mathbb{Z}}^2$ with general dependence axis

We discuss anisotropic scaling of long-range dependent linear random fields $X$ on ${\mathbb{Z}}^2$ with arbitrary dependence axis (direction in the plane along which the moving-average coefficients decay at a smallest rate). The scaling limits are taken over rectangles whose sides are parallel to the coordinate axes and increase as $λ$ and $λ^γ$ when $λ\to \infty$, for any $γ>0$. The scaling transition occurs at $γ^X_0 >0$ if the scaling limits of $X$ are different and do not depend on $γ$ for $γ> γ^X_0$ and $γ< γ^X_0$. We prove that the fact of `oblique' dependence axis (or incongruous scaling) dramatically changes the scaling transition in the above model so that $γ_0^X = 1$ independently of other parameters, contrasting the results in Pilipauskaitė and Surgailis (2017) on the scaling transition under congruous scaling.

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