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 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.