Short and Long Memory Fractional Ornstein–Uhlenbeck α-Stable Processes

Abstract

We show that, similar to the Gaussian case, the fractional Ornstein–Uhlenbeck α-stable process obtained via the Lamperti transformation of the linear fractional stable motion is a different stationary process than the one defined as the solution of the Langevin equation driven by a linear fractional stable noise. We investigate the asymptotic dependence structure of the first process and prove that, in contrast to the second case, it is a short-memory process in the sense of the measure of dependence appropriate for processes with infinite second moment.

Keywords:

• Lamperti transformation
• Langevin equation
• Long memory
• Ornstein–Uhlenbeck process
• Stable processes

Mathematics Subject Classification:

• Primary 60H10
• Secondary 60G52