On the Multivariate Upcrossings Index

Abstract

The multivariate extremal index function is a measure of the clustering among the extreme values of a multivariate stationary sequence. In this article, we introduce a measure of the degree of clustering of upcrossings in a multivariate stationary sequence, called multivariate upcrossings index, which is a multivariate generalization of the concept of upcrossings index. We derive the main properties of this function, namely the relations with the multivariate extremal index and the clustering of upcrossings.

Imposing general local and asymptotic dependence restrictions on the sequence or on its marginals we compute the multivariate upcrossings index from the marginal upcrossings indices and from the joint distribution of a finite number of variables. A couple of illustrative examples are exploited.

Keywords:

• Local dependence conditions
• Multivariate extremal index
• Multivariate upcrossings index
• Point processes of upcrossings
• Stationarity

Mathematics Subject Classification:

• 60G70

Acknowledgments

We are grateful to the anonymous referee for his detailed comments and suggestions which helped considerably the final form of this article.

This research was supported by the research unit “Centro de Matemática” of the University of Beira Interior and the research project PTDC/MAT/108575/2008 through the Foundation for Science and Technology (FCT) and co-financed by FEDER/COMPETE.