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We study bivariate stochastic recurrence equations (SREs) motivated byapplications to GARCH(1,1) processes. If coefficient matrices of SREs havestrictly positive entries, then the Kesten result applies and it givessolutions with regularly varying tails. Moreover, the tail indices are the samefor all coordinates. However, for applications, this framework is toorestrictive. We study SREs when coefficients are triangular matrices and provethat the coordinates of the solution may exhibit regularly varying tails withdifferent indices. We also specify each tail index together with its constant.The results are used to characterize regular variations of bivariate stationaryGARCH(1,1) processes.

Componentwise different tail solutions for bivariate stochastic recurrence equations -- with application to GARCH(1,1) processes --