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

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, ...} are i.i.d. random matrices. In the case d =1, under a nondegeneracy condition, Goldie and Mailer 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 parallel to Pi(n)(t=1) M-t parallel to -greater than(a.s.) 0 as n -greater than infinity, most of the results of Goldie and Mailer can be extended to the case d greater than 1. If this condition does not hold, some of their results cannot be extended