  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 $\hat Z =
\sum_{i=1}^N A_i Z_i$. 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 $\hat
Z$. If the matrix $\mathbb{E}[N] \mathbb{E} [A]$ 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.
