Tail indices for AX+B Recursion with Triangular Matrices

We study multivariate stochastic recurrence equations (SREs) with triangular matrices. If coefficient matrices of SREs have strictly positive entries, the classical Kesten result says that the stationary solution is regularly varying and the tail indices are the same in all directions. This framewor...

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Bibliographic Details
Published inJournal of theoretical probability Vol. 34; no. 4; pp. 1831 - 1869
Main Authors Matsui, Muneya, Świątkowski, Witold
Format Journal Article
LanguageEnglish
Published New York Springer US 2021
Springer Nature B.V
Subjects
Autoregressive models
Mathematics
Mathematics and Statistics
Probability Theory and Stochastic Processes
Statistics
Stochastic models
Kesten’s theorem
Stochastic recurrence equation
Triangular matrices
Multivariate GARCH(1, 1) processes
60G70
Secondary 62M10
Regular variation
60H25
Primary 60G10
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Summary:We study multivariate stochastic recurrence equations (SREs) with triangular matrices. If coefficient matrices of SREs have strictly positive entries, the classical Kesten result says that the stationary solution is regularly varying and the tail indices are the same in all directions. This framework, however, is too restrictive for applications. In order to widen applicability of SREs, we consider SREs with triangular matrices and we prove that their stationary solutions are regularly varying with component-wise different tail exponents. Several applications to GARCH models are suggested.
ISSN:0894-9840
1572-9230
DOI:10.1007/s10959-020-01019-8