Aggregation of autoregressive random fields and anisotropic long-range dependence

Abstract

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 $(1<\alpha\le2)$ 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” $A=1$.