Fractionally integrated moving average stable processes with long-range dependence

Abstract

Long memory processes driven by Lévy 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 that decays like a power function. Here, we study a class of Lévy processes whose second-order moments are infinite, the so-called α-stable processes. Based on Samorodnitsky and Taqqu (1994), 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, an integration by parts formula follows naturally. We then present a family of stationary SαS processes with the property of long-range dependence, using a generalized measure to investigate its dependence structure. At 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.