On joint weak convergence of partial sum and maxima processes

摘要

For a strictly stationary sequence of random variables we derive functionalconvergence of the joint partial sum and partial maxima processes under jointregular variation with index $lpha in (0,1)$ and weak dependence conditions.The convergence takes place in the space of $mathbb{R}^{2}$-valuedc`{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.