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For a strictly stationary sequence of $\mathbb{R}_{+}^{d}$--valued randomvectors we derive functional convergence of partial maxima stochastic processesunder joint regular variation and weak dependence conditions. The limit processis 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 theSkorohod weak $M_{1}$ topology. We also show that this topology in general cannot be replaced by the stronger (standard) $M_{1}$ topology. The theory isillustrated on three examples, including the multivariate squared GARCH processwith constant conditional correlations.

Weak convergence of multivariate partial maxima processes