Invariance principles for some FARIMA and nonstationary linear processes in the domain of a stable distribution

TL;DR

This paper establishes invariance principles for generalized FARIMA and nonstationary linear processes with stable distribution innovations, extending fractional Lévy processes and introducing a novel sample-splitting technique.

Contribution

It introduces new invariance principles for processes with stable distribution innovations and extends fractional Lévy processes, using a novel sample-splitting analytical technique.

Findings

Proved invariance principles for generalized FARIMA processes

Extended fractional Lévy processes to non-Gaussian stable distributions

Developed a sample-splitting technique applicable to stable distribution processes

Abstract

We prove some invariance principles for processes which generalize FARIMA processes, when the innovations are in the domain of attraction of a nonGaussian stable distribution. The limiting processes are extensions of the fractional L\'evy processes. The technique used is interesting in itself; it extends an older idea of splitting a sample into a central part and an extreme one, analyzing each part with different techniques, and then combining the results. This technique seems to have the potential to be useful in other problems in the domain of nonGaussian stable distributions.