Scaling limits for random fields with long-range dependence

TL;DR

This paper investigates the asymptotic behavior of spatial random fields generated by scattered sets with long-range dependence, revealing different limit processes depending on the scaling of set density and volume.

Contribution

It introduces a comprehensive analysis of the scaling limits of such fields, identifying conditions for Gaussian, independent, and novel non-stable limits.

Findings

Gaussian limits with long-range dependence under fast density growth

Independent scattered limits with infinite second moments under slow density growth

Existence of a non-stable limit in an intermediate scaling regime

Abstract

This paper studies the limits of a spatial random field generated by uniformly scattered random sets, as the density $\lambda$ of the sets grows to infinity and the mean volume $\rho$ of the sets tends to zero. Assuming that the volume distribution has a regularly varying tail with infinite variance, we show that the centered and renormalized random field can have three different limits, depending on the relative speed at which $\lambda$ and $\rho$ are scaled. If $\lambda$ grows much faster than $\rho$ shrinks, the limit is Gaussian with long-range dependence, while in the opposite case, the limit is independently scattered with infinite second moments. In a special intermediate scaling regime, there exists a nontrivial limiting random field that is not stable.