A note on joint functional convergence of partial sum and maxima for linear processes

TL;DR

This paper demonstrates that for linear processes with all coefficients of the same sign, the joint convergence of partial sums and maxima occurs in a stronger topology than previously established, enhancing the understanding of their asymptotic behavior.

Contribution

It establishes joint functional convergence in the Skorohod weak M1 topology for linear processes with same sign coefficients, strengthening prior results that used the M2 topology.

Findings

Convergence in M1 topology under same sign coefficients

Extension of previous results to stronger topology

Improved understanding of joint process behavior

Abstract

Recently, for the joint partial sum and partial maxima processes constructed from linear processes with independent identically distributed innovations that are regularly varying with tail index $\alpha \in (0, 2)$, a functional limit theorem with the Skorohod weak $M_{2}$ topology has been obtained. In this paper we show that, if all the coefficients of the linear processes are of the same sign, the functional convergence holds in the stronger topology, i.e. in the Skorohod weak $M_{1}$ topology on the space of $\mathbb{R}^{2}$--valued c\`{a}dl\`{a}g functions on $[0, 1]$.