Anisotropic scaling limits of long-range dependent random fields

Abstract

We review recent results on anisotropic scaling limits and the scaling transition for linear and their subordinated nonlinear long-range dependent stationary random fields X on ℤ2. The scaling limits $$ {V}_{\upgamma}^X $$ are taken over rectangles in ℤ2 whose sides increase as O(λ) and O(λγ ) as λ→∞for any fixed γ > 0. The scaling transition occurs at $$ {\upgamma}_0^X>0 $$ provided that $$ {V}_{\upgamma}^X $$ are different for $$ \upgamma >{\upgamma}_0^X $$ and $$ \upgamma <{\upgamma}_0^X $$ and do not depend on γ otherwise.