On multidimensional Mandelbrot's cascades

Let $Z$ be a random variable with values in a proper closed convex cone $C\subset {\mathbb{R}}^d$, $A$ a random endomorphism of $C$ and $N$ a random integer. We assume that $Z$, $A$, $N$ are independent. Given $N$ independent copies $(A_i,Z_i)$ of $(A,Z)$ we define a new random variable $\hat Z = \sum_{i=1}^N A_i Z_i$\mathrm{.}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 $\hat Z$\mathrm{.}Ifthematrix$$\mathbb{E}$[N] $\mathbb{E}$ [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.