Title: Componentwise different tail solutions for bivariate stochastic recurrence equations -- with application to GARCH(1,1) processes --,
Abstract: We study bivariate stochastic recurrence equations (SREs) motivated by
applications to GARCH(1,1) processes. If coefficient matrices of SREs have
strictly positive entries, then the Kesten result applies and it gives
solutions with regularly varying tails. Moreover, the tail indices are the same
for all coordinates. However, for applications, this framework is too
restrictive. We study SREs when coefficients are triangular matrices and prove
that the coordinates of the solution may exhibit regularly varying tails with
different indices. We also specify each tail index together with its constant.
The results are used to characterize regular variations of bivariate stationary
GARCH(1,1) processes.