Self-similar random fields and rescaled random balls model

Abstract. We study generalized random fields which arise as rescaling limits of spa-tial 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 self-similarity properties. Our main result states that all self-similar, translation and rotation invariant Gaussian fields can be obtained through a unified zooming proce-dure 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 L2-representations of the asymptotic generalized random fields viewed as continuous random linear functionals