Scaling limits for random fields with long-range dependence

This paper studies the limits of a spatial random field generated by uniformly scattered random sets, as the density $λ$ of the sets grows to infinity and the mean volume $ρ$ 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 $λ$ and $ρ$ are scaled. If $λ$ grows much faster than $ρ$ 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.

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