  We provide new, mild conditions for strict stationarity and ergodicity of a
class of BEKK processes. By exploiting that the processes can be represented as
multivariate stochastic recurrence equations, we characterize the tail behavior
of the associated stationary laws. Specifically, we show that the each
component of the BEKK processes is regularly varying with some tail index. In
general, the tail index differs along the components, which contrasts most of
the existing literature on the tail behavior of multivariate GARCH processes.
