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 $O(\mathit{\lambda })$ and $O({\mathit{\lambda }}^{\mathit{\gamma }})$ respectively as $\mathit{\lambda }\downarrow 0$, for some $\mathit{\gamma }>0$. We consider two types of increments of X: usual increment and rectangular increment, leading to the respective concepts of γ-tangent and γ-rectangent random fields. We prove that for above X both types of local scaling limits exist for any $\mathit{\gamma }>0$ and undergo a transition, being independent of $\mathit{\gamma }>{\mathit{\gamma }}_0$ and $\mathit{\gamma }<{\mathit{\gamma }}_0$, for some ${\mathit{\gamma }}_0>0$; moreover, the ‘unbalanced’ scaling limits ($\mathit{\gamma }\ne {\mathit{\gamma }}_0$) are $(H_1,H_2)$-multi self-similar with one of $H_i$, $i=1,2$, equal to 0 or 1. The paper extends Pilipauskait˙e and Surgailis (Stochastic Process. Appl. 127 (2017) 2751–2779) and Surgailis (Stochastic Process. Appl. 130 (2020) 7518–7546) on large-scale anisotropic scaling of random fields on ${\mathbb{Z}}^2$ and Benassi et al. (Bernoulli 10 (2004) 357–373) on 1-tangent limits of isotropic fractional Lévy random fields.