Scaling Limits of Linear Random Fields on ${\mathbb {Z}}^2$ with General Dependence Axis

Abstract

We discuss anisotropic scaling limits 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 $V^X_{\gamma }$ are random fields on $\mathbb {R}^2_+$ defined as the limits (in the sense of finite-dimensional distributions) of partial sums of X taken over rectangles with sides increasing along horizontal and vertical directions at rates λ and λ ^γ respectively as λ →∞ for arbitrary fixed γ > 0. The scaling limits generally depend on γ and constitute a one-dimensional family $\{V^X_{\gamma }, \gamma >0\}$ of random fields. The scaling transition occurs at some $\gamma ^X_0 >0$ if $V^X_\gamma$ are different and do not depend on γ 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 Pilipauskaitė and Surgailis (2017) on the scaling transition under congruous scaling.

To the memory of Vladas and the time we enjoyed together in Vilnius, Rio and Shanghai