Joint functional convergence of partial sum and maxima for linear processes

TL;DR

This paper establishes joint functional limit theorems for partial sums and maxima of linear processes with heavy-tailed innovations, expanding understanding of their asymptotic behavior in specific function spaces.

Contribution

It introduces new joint convergence results for partial sums and maxima of linear processes with heavy tails, under specific coefficient conditions.

Findings

Functional convergence in Skorohod M2 topology for joint processes

Joint convergence in M2 and M1 topologies for sum and maxima

Extension of limit theorems to processes with heavy-tailed innovations

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.