On discrete-time stable multiple Markov processes

ABSTRACT

In this paper, we provide necessary conditions for discrete-time symmetric α-stable processes to be linear 2-ple Markov. The aim of this paper is to extend the results given by Adler et al. (1990) to general multiple Markov processes, called linear multiple Markov processes. A necessary and sufficient condition based on the covariation for SαS processes to be linear multiple Markov is provided. A complete description of this class of covariation functions including the stationary case is given. We show that the sum of two independent time-changed Levy motions is linear multiple Markov of order 2. The SαS conditional distributions are also studied, and the conditional characteristic functions of linear 2-ple Markov SαS processes are formed. Finally, we study two famous classes of SαS processes, namely, sub-Gaussian and harmonizable processes, and discuss their Markov properties.

KEYWORDS:

• Markov stable processes
• second-order linear homogeneous recurrence
• stable Ornstein Uhlenbeck processes
• sub-Gaussian and harmonizable processes
• time-changed levy motion

MATHEMATICS SUBJECT CLASSIFICATION:

• 60J25
• 60G10

Acknowledgments

The authors would like to thank a referee for his/her constructive comments and suggestions. This work was supported by the Research Council of Shiraz University.