On the mixing structure of stationary increment and self-similar symmetric \alpha-stable processes

TL;DR

This paper characterizes stationary increment and self-similar symmetric -stable processes using nonsingular flows and cocycles, providing a structural understanding of their mixing properties and examples.

Contribution

It offers a novel characterization of these processes through flow theory, linking their structure to ergodic components and expanding the understanding of mixed moving average processes.

Findings

Characterization of processes via nonsingular flows and cocycles

Examples of self-similar mixed moving average processes

Insight into the ergodic decomposition of SS processes

Abstract

Mixed moving average processes appear in the ergodic decomposition of stationary symmetric \alpha-stable (S\alpha S) processes. They correspond to the dissipative part of "deterministic" flows generating S\alpha S processes (Rosinski, 1995). Along these lines we study stationary increment and self-similar S\alpha S processes. Since the classes of stationary increment and self-similar processes can be embedded into the class of stationary processes by the Masani and Lamperti transformations, respectively, we characterize these classes of S\alpha S processes in terms of nonsingular flows and the related cocycles. We illustrate this approach considering various examples of self-similar mixed moving average S\alpha S processes introduced in (Surgailis, Rosinski, Mandrekar and Cambanis, 1992).