  A $d$-dimensional RCA(1) process is a generalization of the $d$-dimensional
AR(1) process, such that the coefficients $\{M_t;t=1,2,\ldots\}$ are i.i.d.
random matrices. In the case $d=1$, under a nondegeneracy condition, Goldie and
Maller gave necessary and sufficient conditions for the convergence in
distribution of an RCA(1) process, and for the almost sure convergence of a
closely related sum of random variables called a perpetuity. We here prove that
under the condition $\Vert {\prod_{t=1}^nM_t}\Vert
\stackrel{\mathrm{a.s.}}{\longrightarrow}0$ as $n\to\infty$, most of the
results of Goldie and Maller can be extended to the case $d>1$. If this
condition does not hold, some of their results cannot be extended.
