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 ${\gamma }_0^X>0$ if the scaling limits of $X$ are different and do not depend on $\gamma$ for $\gamma >{\gamma }_0^X$ and $\gamma <{\gamma }_0^X$. 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 Pilipauskaitė and Surgailis (2017) on the scaling transition under congruous scaling.

References in corpus (4)

• Central limit theorems for long range dependent spatial linear processes
• Scaling transition for nonlinear random fields with long-range dependence
• Anisotropic scaling of random grain model with application to network traffic
• Some remarks on scaling transition in limit theorems for random fields