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

## TL;DR

This paper investigates bivariate stochastic recurrence equations with triangular coefficient matrices, revealing that solution coordinates can have different tail behaviors, and applies findings to characterize GARCH(1,1) processes.

## Contribution

It extends the analysis of SREs to cases with triangular matrices, showing different tail indices for each coordinate, unlike the classical positive matrix case.

## Key findings

- Coordinates can have different tail indices.
- Explicit tail constants are derived.
- Results apply to bivariate 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.