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International audienc

Let Z be a random variable with values in a proper closed convex cone C ⊂ R d , A a random endomorphism of C and N a random integer. We assume that Z, A, N are independent.Let T be the corresponding transformation on the set of probability measures on C i.e. T maps the law of Z to the law of Z. If the matrix E[N ]E[A] has dominant eigenvalue 1, we study existence and properties of fixed points of T having finite nonzero expectation. Existing one dimensional results concerning T are extended to higher dimensions. In particular we give conditions under which such fixed points of T have multidimensional regular variation in the sense of extreme value theory and we determine the index of regular variation.