SOME LIMIT-THEOREMS FOR EXTREMAL AND UNION SHOT-NOISE PROCESSES

We study the limiting behaviour of suitably normalized union shot-noise processes [GRAPHICS] t is-an-element-of R(d), where F is a set-valued function on R(d) x M, {beta(i), i greater-than-or-equal-to 1} is a sequence of i.i.d. random elements on some measurable space [M, M] and PSI = {x(i), i greater-than-or-equal-to 1} stands for a stationary d-dimensional point process whose intensity lambda tends to infinity. General results concerning weak convergence of parametrized union shot-noise processes XI(epsilon)(t) as epsilon down 0 are obtained (Theorem 5.1 and its corollaries), if the point process lambda(1/d)PSI 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.