Local scaling limits of Lévy driven fractional random fields

Abstract

We obtain a complete description of local anisotropic scaling limits for a class of fractional random fields $X$ on ${\mathbb{R}}^2$ written as stochastic integral with respect to infinitely divisible random measure. The scaling procedure involves increments of $X$ over points the distance between which in the horizontal and vertical directions shrinks as $O(\lambda )$ and $O({\lambda }^{\gamma })$ respectively as $\lambda \downarrow 0$, for some $\gamma >0$. We consider two types of increments of $X$: usual increment and rectangular increment, leading to the respective concepts of $\gamma$-tangent and $\gamma$-rectangent random fields. We prove that for above $X$ both types of local scaling limits exist for any $\gamma >0$ and undergo a transition, being independent of $\gamma >{\gamma }_0$ and $\gamma <{\gamma }_0$, for some ${\gamma }_0>0$; moreover, the "unbalanced" scaling limits ($\gamma \ne {\gamma }_0$) are $(H_1,H_2)$-multi self-similar with one of $H_i$, $i=1,2$, equal to $0$ or $1$. The paper extends Pilipauskaitė and Surgailis (2017) and Surgailis (2020) on large-scale anisotropic scaling of random fields on ${\mathbb{Z}}^2$ and Benassi et al. (2004) on $1$-tangent limits of isotropic fractional Lévy random fields.

Accepted for publication in Bernoulli

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