Self-similar Random Fields and Rescaled Random Balls Models

Abstract

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 renormalized random balls field admits a limit with self-similarity properties. Our main result states that all self-similar, translation- and rotation-invariant Gaussian fields can be obtained through a unified zooming procedure starting from a random balls model. This approach has to be understood as a microscopic description of macroscopic properties. Under specific assumptions, we also get a Poisson-type asymptotic field. In addition to investigating stationarity and self-similarity properties, we give L 2-representations of the asymptotic generalized random fields viewed as continuous random linear functionals.