Conditions for convergence of random coefficient $\mathrm{AR}(1)$ processes and perpetuities in higher dimensions

Abstract

A $d$-dimensional $\mathrm{RCA}(1)$ process is a generalization of the $d$-dimensional $\mathrm{AR}(1)$ 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 $\mathrm{RCA}(1)$ 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 $\Vert{\prod_{t=1}^{n}M_{t}}\Vert\stackrel{\mathrm{a.s.}}{\longrightarrow}0$ 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.