On the tail behavior of a class of multivariate conditionally heteroskedastic processes
Conditions for geometric ergodicity of multivariate autoregressive conditional heteroskedasticity (ARCH) processes, with the so-called BEKK (Baba, Engle, Kraft, and Kroner) parametrization, are considered. We show for a class of BEKK-ARCH processes that the invariant distribution is regularly varying. In order to account for the possibility of different tail indices of the marginals, we consider the notion of vector scaling regular variation (VSRV), closely related to non-standard regular variation. The characterization of the tail behavior of the processes is used for deriving the asymptotic properties of the sample covariance matrices.
KeywordsStochastic recurrence equations Markov processes Regular variation Multivariate ARCH Asymptotic properties Geometric ergodicity
AMS 2000 Subject Classifications60G70 60G10 60H25 39A50
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We are grateful for comments and suggestions from the editor-in-chief (Thomas Mikosch), an associate editor, and two referees, which have led to a much improved manuscript. Moreover, we thank Sebastian Mentemeier for valuable comments. Pedersen greatly acknowledges funding from the Carlsberg Foundation. Financial support by the ANR network AMERISKA ANR 14 CE20 0006 01 is gratefully acknowledged by Wintenberger.
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