  A central issue in the theory of extreme values focuses on suitable
conditions such that the well-known results for the limiting distributions of
the maximum of i.i.d. sequences can be applied to stationary ones. In this
context, the extremal index appears as a key parameter to capture the effect of
temporal dependence on the limiting distribution of the maxima. The
multivariate extremal index corresponds to a generalization of this concept to
a multivariate context and affects the tail dependence structure within the
marginal sequences and between them. As it is a function, the inference becomes
more difficult, and it is therefore important to obtain characterizations,
namely bounds based on the marginal dependence that are easier to estimate. In
this work we present two decompositions that emphasize different types of
information contained in the multivariate extremal index, an upper limit better
than those found in the literature and we analyze its role in dependence on the
limiting model of the componentwise maxima of a stationary sequence. We will
illustrate the results with examples of recognized interest in applications.
