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

We study the limiting behaviour of suitably normalized union shot-nois e processes [GRAPHICS] t is-an-element-of R(d), where F is a set-value d function on R(d) x M, {beta(i), i greater-than-or-equal-to 1} is a s equence 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-d imensional point process whose intensity lambda tends to infinity. Gen eral results concerning weak convergence of parametrized union shot-no ise 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 wea k limit and F satisfies some technical conditions. An essential tool f or proving these results is the notion of regular variation of multiva lued functions. Some examples illustrate the applicability of the resu lts.