On multidimensional Mandelbrot cascades

Abstract

Let Z be a random variable with values in a proper closed convex cone , A a random endomorphism of C and N a random integer. We assume that Z, A, N are independent. Given N independent copies of we define a new random variable . 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 . If the matrix has dominant eigenvalue 1, we study existence and properties of fixed points of T having finite non-zero 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.