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A $d$-dimensional RCA(1) process is a generalization of the $d$-dimensionalAR(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 andMaller gave necessary and sufficient conditions for the convergence indistribution of an RCA(1) process, and for the almost sure convergence of aclosely related sum of random variables called a perpetuity. We here prove thatunder the condition $\Vert {\prod_{t=1}^nM_t}\Vert\stackrel{\mathrm{a.s.}}{\longrightarrow}0$ as $n\to\infty$, most of theresults of Goldie and Maller can be extended to the case $d>1$. If thiscondition does not hold, some of their results cannot be extended.

Conditions for convergence of random coefficient AR(1) processes and perpetuities in higher dimensions