Skorokhod $M_{1}$ convergence of maxima of multivariate linear processes with heavy-tailed innovations and random coefficients

TL;DR

This paper establishes the convergence of the maximum values in multivariate linear processes with heavy-tailed innovations and random coefficients, using the weak Skorokhod M1 topology, highlighting its necessity over the standard M1 topology.

Contribution

It provides the first functional convergence results for maxima of such processes in the weak Skorokhod M1 topology, emphasizing the importance of this topology in heavy-tailed, dependent settings.

Findings

Convergence in the weak Skorokhod M1 topology for multivariate maxima.

Standard M1 topology is insufficient for this convergence.

Heavy-tailed innovations and random coefficients are key factors in the analysis.

Abstract

We derive functional convergence of the partial maxima stochastic processes of multivariate linear processes with weakly dependent heavy-tailed innovations and random coefficients. The convergence takes place in the space of $\mathbb{R}^{d}$--valued c\`{a}dl\`{a}g functions on $[0,1]$ endowed with the weak Skorokhod $M_{1}$ topology. We also show that this topology in general can not be replaced by the standard (or strong) $M_{1}$ topology.