Tail indices for AX+B recursion with triangular matrices

2018/08/29 by Muneya Matsui, Matsui, Muneya, W. Świa̧tkowski +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #Financial Risk and Volatility Modeling #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1808.09678

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.

Citations

• Random difference equations and Renewal theory for products of random matrices

Cited by

• Characterization of the tail behavior of a class of BEKK processes: A stochastic recurrence equation approach

Related

• Tails of bivariate stochastic recurrence equation with triangular matrices
• Componentwise different tail solutions for bivariate stochastic recurrence equations -- with application to GARCH(1,1) processes --
• Stochastic recurrence equation with diagonal matrices
• Hidden regular variation for stochastic recursions with diagonal matrices
• Analysing heavy-tail properties of Stochastic Gradient Descent by means of Stochastic Recurrence Equations