On the multivariate upcrossings index

Abstract

The notion of multivariate upcrossings index of a stationary sequence $\mathbf{X}=\{(X_{n,1},\dots ,X_{n,d}){\}}_{n\ge 1}$ is introduced and its main properties are derived, namely the relations with the multivariate extremal index and the clustering of upcrossings. Under asymptotic independence conditions on the marginal sequences of $\mathbf{X}$ the multivariate upcrossings index is obtained from the marginal upcrossings indices. For a class of stationary multidimensional sequences assumed to satisfy a mild oscillation restriction, the multivariate upcrossings index is computed from the joint distribution of a finite number of variables. The upcrossings index is calculated for two examples of bivariate sequences.

20 pages