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

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 $λ$ and $λ^γ$, for any fixed $γ>0$. We prove that for above $X$ the partial sums or scaling limits exist for any $γ>0$ and exhibit a scaling transition at some $γ= γ_0>0$; moreover, the `unbalanced' scaling limits ($γ\neγ_0$) 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.

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