### Abstract

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

*Annals of Probability*,

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

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*Annals of Probability*, vol. 35, no. 2, pp. 528-550. https://doi.org/10.1214/009117906000000700

**Scaling limits for random fields with long-range dependence.** / Kaj, Ingemar; Leskelä, Lasse; Norros, Ilkka; Schmidt, Volker.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Scaling limits for random fields with long-range dependence

AU - Kaj, Ingemar

AU - Leskelä, Lasse

AU - Norros, Ilkka

AU - Schmidt, Volker

PY - 2007

Y1 - 2007

N2 - 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.

AB - 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.

U2 - 10.1214/009117906000000700

DO - 10.1214/009117906000000700

M3 - Article

VL - 35

SP - 528

EP - 550

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 2

ER -