  We consider a generalization of the weighted random ball model. The model is
driven by a random Poisson measure with a product heavy tailed intensity
measure. Such a model typically represents the transmission of a network of
stations with a fading effect. In a previous article, the authors proved the
convergence of the finite-dimensional distributions of related generalized
random fields under various scalings and in the particular case when the fading
function is the indicator function of the unit ball. In this paper, tightness
and functional convergence are investigated. Using suitable moment estimates,
we prove functional convergences for some parametric classes of configurations
under the so-called large ball scaling and intermediate ball scaling.
Convergence in the space of distributions is also discussed.
