Rescaled weighted random ball models and stable self-similar random fields

We consider weighted random balls in distributed according to a random Poisson measure with heavy-tailed intensity and study the asymptotic behavior of the total weight of some configurations in while we perform a zooming operation. The resulting procedure is very rich and several regimes appear in the limit, depending on the intensity of the balls, the zooming factor, the tail parameters of the radii and the weights. Statistical properties of the limit fields are also evidenced, such as isotropy, self-similarity or dependence. One regime is of particular interest and yields [alpha]-stable stationary isotropic self-similar generalized random fields which recovers Takenaka fields, Telecom process or fractional Brownian motion.