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

Abstract

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 $\lambda \to \infty$, for any $\gamma >0$. The scaling transition occurs at $\gamma^X_0 >0$ if the scaling limits of $X$ are different and do not depend on $\gamma$ 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.