Functional Convergence of Linear Processes with Heavy-Tailed Innovations

TL;DR

This paper investigates the convergence behavior of partial sums of linear processes with heavy-tailed innovations, providing new conditions for convergence to stable Lévy motions and extending existing functional convergence results.

Contribution

It offers necessary and sufficient conditions for finite-dimensional convergence to $ ext{alpha}$-stable Lévy motions and extends functional convergence results to the $S$ topology.

Findings

Established conditions for convergence to $ ext{alpha}$-stable Lévy motions.

Extended functional convergence results to the $S$ topology.

Provided new, tractable conditions for $ ext{alpha} \, ext{≤} \, 1$.

Abstract

We study convergence in law of partial sums of linear processes with heavy-tailed innovations. In the case of summable coefficients necessary and sufficient conditions for the finite dimensional convergence to an $\alpha$-stable L\'evy Motion are given. The conditions lead to new, tractable sufficient conditions in the case $\alpha \leq 1$. In the functional setting we complement the existing results on $M_1$-convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (1992) and improved by Louhichi and Rio (2011), by proving that in the general setting partial sums of linear processes are convergent on the Skorokhod space equipped with the $S$ topology, introduced by Jakubowski (1997).