  Multivariate stochastic recurrence equations (SREs) are investigated when
coefficients are triangular matrices. If coefficient matrices of SREs have all
strictly positive elements, the Kesten's classical result yields solutions with
regularly varying tails such that the tail indices of solutions are the same
through coordinates. This framework is too restrictive for applications. In
order to widen the applicability of the SREs, we study SREs with triangular
matrix coefficients and prove that they have regularly varying solutions which
may exhibit coordinate-wisely different tail exponents. We also specify the
coefficients for regularly varying tails. Several applications are suggested
for GARCH models.
