Correlation properties of continuous-time autoregressive processes delayed by the inverse of the stable subordinator

Abstract

We define the delayed Lévy-driven continuous-time autoregressive process via the inverse of the stable subordinator. We derive correlation structure for the observed non-stationary delayed Lévy-driven continuous-time autoregressive processes of order p, emphasizing low orders, and we show they exhibit long-range dependence property. Distributional properties are discussed as well.

Keywords:

• Continuous-time autoregressive process
• Lévy noise
• delayed stochastic process
• inverse of the stable subordinator
• Mittag-Leffler function
• correlation structure

Mathematics Subject Classification (2010):

• 37M10
• 62M10
• 60G51

Acknowledgments

N. Leonenko was supported in particular by Cardiff Incoming Visiting Fellowship Scheme and International Collaboration Seedcorn Fund, Cardiff Data Innovation Research Institute Seed Corn Funding, Australian Research Council’s Discovery Projects funding scheme (project number DP160101366), and by projects MTM2012-32674 and MTM2015–71839–P (co-funded with Federal funds), of the DGI, MINECO, Spain.

I. Papić was supported by the project number IZIP-2016-22 funded by the J. J. Strossmayer University of Osijek.