Scaling limits for random fields with long-range dependence

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

2007-07-25

arxiv.org/abs/0707.3729v1

Annals of Probability 2007, Vol. 35, No. 2, 528-550

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117906000000700 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117906000000700

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.

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