Let $Z$ be a random variable with values in a proper closed convex cone $C\subset \R^d$, $A$ a random endomorphism of $C$ and $N$ a random integer. Given $N$ independent copies $(A_i,Z_i)$ of $(A,Z)$ we define a new random variable $\wh Z = \sum_{i=1}^N A_i Z_i$. Let $T$ be the corresponding transformation on the set of probability measures on $C$. We study existence and properties of fixed points of $T$. Previous one dimensional results on existence of fixed points of T as well as on homogeneity of their tails are extended to higher dimensions.
