Weak convergence of multivariate partial maxima processes

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.