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Anisotropic scaling limits of long-range dependent random fields. (English) Zbl 1456.60123
Summary: 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 $$\mathbb{Z}^2$$. The scaling limits $${V}_{\gamma}^X$$ are taken over rectangles in $$\mathbb{Z}^2$$ whose sides increase as $$O( \lambda )$$ and $$O(\lambda \gamma$$ ) as $$\lambda \rightarrow \infty$$ for any fixed $$\gamma > 0$$. The scaling transition occurs at $${\gamma}_0^X>0$$ provided that $${V}_{\gamma}^X$$ are different for $$\gamma >{\gamma}_0^X$$ and $$\gamma <{\gamma}_0^X$$ and do not depend on $$\gamma$$ otherwise.

##### MSC:
 60G60 Random fields 60G15 Gaussian processes 60G18 Self-similar stochastic processes 60G22 Fractional processes, including fractional Brownian motion
longmemo
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