Aggregation of autoregressive random fields and anisotropic long-range dependence

We introduce the notions of scaling transition and distributional long-range dependence for stationary random fields $Y$ on ${\mathbb{Z}}^2$ whose normalized partial sums on rectangles with sides growing at rates $O(n)$ and $O(n^{\gamma })$ tend to an operator scaling random field $V_{\gamma }$ on ${\mathbb{R}}^2$, for any $\gamma >0$. The scaling transition is characterized by the fact that there exists a unique ${\gamma }_0>0$ such that the scaling limits $V_{\gamma }$ are different and do not depend on $\gamma$ for $\gamma >{\gamma }_0$ and $\gamma <{\gamma }_0$. The existence of scaling transition together with anisotropic and isotropic distributional long-range dependence properties is demonstrated for a class of $\alpha$-stable $(1<\alpha \le 2)$ aggregated nearest-neighbor autoregressive random fields on ${\mathbb{Z}}^2$ with a scalar random coefficient $A$ having a regularly varying probability density near the “unit root” $A=1$.