Scaling transition for singular linear random fields on $\mathbb{Z}^2$: spectral approach

Abstract

We study partial sums limits of linear random fields $X$ on $\mathbb{Z}^2 $ with spectral density $f({\mathbf x}) $ tending to $\infty,\, 0$ or to both (along different subsequences) as ${\mathbf x} \to (0,0)$. The above behaviors are termed (spectrum) long-range dependence, negative dependence, and long-range negative dependence, respectively, and assume an anisotropic power-law form of $f({\mathbf x})$ near the origin. The partial sums are taken over rectangles whose sides increase as $\lambda$ and $\lambda^\gamma $, for any fixed $\gamma >0$. We prove that for above $X$ the partial sums or scaling limits exist for any $\gamma>0$ and exhibit a scaling transition at some $\gamma = \gamma_0>0$; moreover, the `unbalanced' scaling limits ($\gamma\ne\gamma_0$) are Fractional Brownian Sheet with Hurst parameters taking values from $[0,1]$. The paper extends \cite{ps2015, pils2017, sur2020} to the above spectrum dependence conditions and/or more general values of Hurst parameters.