### Abstract

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.

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.

Original language | English |
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Pages (from-to) | 528-550 |

Journal | Annals of Probability |

Volume | 35 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2007 |

MoE publication type | A1 Journal article-refereed |

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### Cite this

Kaj, I., Leskelä, L., Norros, I., & Schmidt, V. (2007). Scaling limits for random fields with long-range dependence.

*Annals of Probability*,*35*(2), 528-550. https://doi.org/10.1214/009117906000000700