Self-similar Random Fields and Rescaled Random Balls Models

We study generalized random fields which arise as rescaling limits of spatial configurations of uniformly scattered random balls as the mean radius of the balls tends to 0 or infinity. Assuming that the radius distribution has a power law behavior, we prove that the centered and re-normalized random balls field admits a limit with spatial dependence and self-similarity properties. In particular, our approach provides a unified framework to obtain all self-similar, translation and rotation invariant Gaussian fields. Under specific assumptions, we also get a Poisson type asymptotic field. In addition to investigating stationarity and self-similarity properties, we give L2-representations of the asymptotic generalized random fields viewed as continuous random linear functionals. Introduction In this work we construct essentially all Gaussian, translation and rotation invariant, H-self-similar generalized random fields on Rd in a unified manner as scaling limits of a random balls model. The self-similarity index H ranges over all of R \ Z and the random balls model is of germ-grain type. It arises by aggregation of spherical grains attached to uniformly scattered germs given by a Poisson point process in d-dimensional space. By a similar scaling procedure, we obtain also non-Gaussian random fields with interesting properties, in particular a model of the type ”fractional Poisson field”. Its covariance functional coincides with that of the Gaussian H-self-similar field, so that it fulfills a second order self-similarity property. Although not self-similar in law, this Poisson field presents a property of ”aggregate similarity” which takes into account both Poisson structure and self-similarity. We observe two distinctly separate behaviors depending on whether the self-similarity index H belongs to an interval of type (m,m + 1/2) or of type [m − 1/2,m) for some integer m. In the first case, the scaling limit applies to random balls models with balls of arbitrarily small radii. The asymptotic field then provides spatial dependence of negative type. In the opposite case, the corresponding germ-grain models have arbitrarily large spherical grains which lead to spatial long range dependence. The scaling procedure which acts on the random balls model is based on the assumption that the grains have random radius, independent and identically distributed, with a distribution having a power law behavior either in zero or at infinity. The resulting 2000 Mathematics Subject Classification. Primary: 60G60, 60G78; Secondary: 60G20, 60D05, 60G55, 60G10, 60F05.