Conditions for geometric ergodicity of multivariate ARCH processes, with the so-called BEKK 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, in the spirit of Perfekt (1997). The characterization of the tail behavior of the processes is used for deriving the asymptotic distribution of the sample covariance matrices.
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