On the multivariate upcrossings index

TL;DR

This paper introduces the multivariate upcrossings index for stationary sequences, explores its properties, and provides methods for its computation, including under asymptotic independence and oscillation restrictions, with practical examples.

Contribution

It defines and analyzes the multivariate upcrossings index, relating it to extremal indices and clustering, and offers new computation techniques for specific classes of sequences.

Findings

The multivariate upcrossings index is related to the extremal index and clustering.

Under asymptotic independence, it can be derived from marginal indices.

Explicit calculations are provided for certain bivariate sequences.

Abstract

The notion of multivariate upcrossings index of a stationary sequence ${\bf{X}}=\{(X_{n,1},\ldots,X_{n,d})\}_{n\geq 1}$ 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 ${\bf{X}}$ 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.