Weak convergence of multivariate partial maxima processes

TL;DR

This paper establishes the weak convergence of multivariate partial maxima processes for stationary sequences under regular variation and weak dependence, with the limit being an extremal process in the Skorohod weak M1 topology.

Contribution

It provides the first functional convergence results for multivariate partial maxima processes under joint regular variation and weak dependence, highlighting the role of the Skorohod weak M1 topology.

Findings

Convergence in the space of càdlàg functions with weak M1 topology.

The strong M1 topology cannot generally replace the weak M1 topology.

Application to multivariate squared GARCH processes with constant correlations.

Abstract

For a strictly stationary sequence of $\mathbb{R}_{+}^{d}$--valued random vectors we derive functional convergence of partial maxima stochastic processes under joint regular variation and weak dependence conditions. The limit process is an extremal process and the convergence takes place in the space of $\mathbb{R}_{+}^{d}$--valued c\`{a}dl\`{a}g functions on $[0,1]$, with the Skorohod weak $M_{1}$ topology. We also show that this topology in general can not be replaced by the stronger (standard) $M_{1}$ topology. The theory is illustrated on three examples, including the multivariate squared GARCH process with constant conditional correlations.