Some Limit Theorems for Extremal and Union Shot-Noise Processes

We study the limiting behaviour of suitably normalized union shot-noise processes , where F is a set-valued function on R^d × ?? is a sequence of i.i.d. random elements on some measurable space [?? ??] and Ψ = {x_i, i≥ 1} stands for a stationary d-dimensional point process whose intensity λ tends to infinity. General results concerning weak convergence of parametrized union shot-noise processes Ξ?(t) as ? ↓ 0 are obtained (Theorem 5.1 and its corollaries), if the point process λ^{1 d}Ψ has a weak limit and F satisfies some technical conditions. An essential tool for proving these results is the notion of regular variation of multivalued functions. Some examples illustrate the applicability of the results.