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 strong spatial
dependence. In particular, our approach provides a unified framework
to obtain all self-similar, translation and rotation
invariant Gaussian fields.
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.
2007. , 26 p.