Fractionally Integrated Moving Average Stable Processes With Long-Range Dependence

TL;DR

This paper introduces a new class of long-range dependent stable processes driven by Lévy noise with infinite second moments, using fractional integrals and isometries to analyze their dependence and law of large numbers.

Contribution

It constructs a novel framework for defining and analyzing long-range dependent stable processes with infinite variance using fractional calculus and measure theory.

Findings

Established a stochastic integral framework for stable processes with infinite variance.

Derived an integration by parts formula for these processes.

Proved a law of large numbers for the constructed processes.

Abstract

Long memory processes driven by L\'evy noise with finite second-order moments have been well studied in the literature. They form a very rich class of processes presenting an autocovariance function which decays like a power function. Here, we study a class of L\'evy process whose second-order moments are infinite, the so-called $\alpha$-stable processes. Based on Samorodnitsky and Taqqu (2000), we construct an isometry that allows us to define stochastic integrals concerning the linear fractional stable motion using Riemann-Liouville fractional integrals. With this construction, follows naturally an integration by parts formula. We then present a family of stationary $S\alpha S$ processes with the property of long-range dependence, using a generalized measure to investigate its dependence structure. In the end, the law of large number's result for a time's sample of the process is shown as an application of the isometry and integration by parts formula.