# Dissecting the multivariate extremal index and tail dependence

## TL;DR

This paper investigates the multivariate extremal index, a key parameter in extreme value theory, providing new decompositions, bounds, and insights into its role in tail dependence and maxima of stationary sequences.

## Contribution

It introduces two novel decompositions of the multivariate extremal index, establishes improved bounds, and analyzes its influence on tail dependence in stationary sequences.

## Key findings

- New decompositions of the multivariate extremal index
- Improved bounds for the extremal index based on marginal dependence
- Analysis of the extremal index's role in the limiting distribution of maxima

## Abstract

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.