TY - JOUR

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

AU - Kaj, I.

AU - Leskelä, L.

AU - Norros, I.

AU - Schmidt, V.

PY - 2007

Y1 - 2007

N2 - This paper studies the limits of a spatial random field generated by uniformly scattered random sets, as the density ¿ of the sets grows to infinity and the mean volume ¿ 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 ¿ and ¿ are scaled. If ¿ grows much faster than ¿ 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 ¿ of the sets grows to infinity and the mean volume ¿ 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 ¿ and ¿ are scaled. If ¿ grows much faster than ¿ 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

SN - 0091-1798

VL - 35

SP - 528

EP - 550

JO - The Annals of Probability

JF - The Annals of Probability

IS - 2

ER -