For a strictly stationary sequence of random variables we derive functional convergence of the joint partial sum and partial maxima process under joint regular variation with index $\alpha \in (0,2)$ and weak dependence conditions. The limiting process consists of an $\alpha$--stable L\'{ e}vy process and an extremal process. We also describe the dependence between these two components of the limit. The convergence takes place in the space of $\mathbb{ R}^{ 2}$--valued c\`{ a}dl\`{ a}g functions on $[0,1]$, with the Skorohod weak $M_{ 1}$ topology. We further show that this topology in general can not be replaced by the stronger (standard) $M_{ 1}$ topology.
Categories: math.PR