, Volume 21, Issue 2, pp 261–284 | Cite as

On the tail behavior of a class of multivariate conditionally heteroskedastic processes

  • Rasmus Søndergaard Pedersen
  • Olivier Wintenberger


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.


Stochastic recurrence equations Markov processes Regular variation Multivariate ARCH Asymptotic properties Geometric ergodicity 

AMS 2000 Subject Classifications

60G70 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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Authors and Affiliations

  1. 1.Department of EconomicsUniversity of CopenhagenCopenhagen KDenmark
  2. 2.Department of Mathematical SciencesUniversity of CopenhagenCopenhagen ØDenmark
  3. 3.Université Pierre et Marie Curie, LSTAParisFrance

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