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 $\lambda$ and ${\lambda }^{\gamma }$ when $\lambda \to \infty$,forany$\gamma >0$\mathrm{.}Thescalingtransitionoccursat$\gamma^X_0

0$ifthescalinglimitsof$X$aredifferentanddonotdependon$\gamma$for$\gamma > \gamma^X_0 $and$\gamma < \gamma^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 ${\gamma }_0^X=1$ independently of other parameters, contrasting the results in Pilipauskait.e and Surgailis (2017) on the scaling transition under congruous scaling.