Tail indices for AX+B recursion with triangular matrices
Abstract
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.
Cite
@article{arxiv.1808.09678,
title = {Tail indices for AX+B recursion with triangular matrices},
author = {Muneya Matsui and Witold Świątkowski},
journal= {arXiv preprint arXiv:1808.09678},
year = {2020}
}
Comments
30 pages, accepted in journal of theoretical probability, Key words: Stochastic recurrence equation, Kesten's theorem, regular variation, multivariate GARCH(1,1) processes, triangular matrices,