Max-geometric infinite divisibility and stability

The authors consider a stability property for R^d -valued random vectors appropriate for describing extreme events up to the time of a catastrophe. Let be geometrically distributed. The random vector is max-geometrically infinitely divisible if for some iid random vectors independent of N(p) we have , for any 0<p<1. Y is max-geometrically stable if for 0<p>1, for , , iid and independent of , we have and of the same type. These distributions are characterized and domains of attraction and related rates of convergence questions explored.