Joint functional convergence of partial sum and maxima for linear processes

Abstract

For linear processes with independent identically distributed innovations that are regularly varying with tail index $\alpha \in (0, 2)$, we study functional convergence of the joint partial sum and partial maxima processes. We derive a functional limit theorem under certain assumptions on the coefficients of the linear processes which enable the functional convergence to hold in the space of $\mathbb{R}^{2}$--valued c\`adl\`ag functions on $[0, 1]$ with the Skorohod weak $M_{2}$ topology. Also a joint convergence in the $M_{2}$ topology on the first coordinate and in the $M_{1}$ topology on the second coordinate is obtained.