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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
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longmemo
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