Abstract
Stable non-Gaussian self-similar mixed moving averages can be decomposed into several components. Two of these are the periodic and cyclic fractional stable motions which are the subject of this study. We focus on the structure of their integral representations and show that the periodic fractional stable motions have, in fact, a canonical representation. We study several examples and discuss questions of uniqueness, namely how to determine whether two given integral representations of periodic or cyclic fractional stable motions give rise to the same process.
Highlights
Periodic and cyclic fractional stable motions (PFSMs and CFSMs, in short) were introduced by Pipiras and Taqqu (2004b) in the context of a decomposition of symmetric α-stable (SαS, in short), α ∈ (0, 2), self-similar processes Xα(t), t ∈ R, with stationary increments having a mixed moving average representation {Xα(t)}t∈R =dG(x, t + u) − G(x, u) Mα(dx, du), XR t∈R (1.1)where =d stands for the equality of the finite-dimensional distributions
We show that PFSMs can be defined as those self-similar mixed moving averages having the representation (1.1) with
Our goal is to study integral representations of PFSMs and CFSMs
Summary
Periodic and cyclic fractional stable motions (PFSMs and CFSMs, in short) were introduced by Pipiras and Taqqu (2004b) in the context of a decomposition of symmetric α-stable (SαS, in short), α ∈ (0, 2), self-similar processes Xα(t), t ∈ R, with stationary increments having a mixed moving average representation. One can use the characteristics of the flows to classify the corresponding H-sssi processes as well as to decompose a given H-sssi process into sub-processes that belong to disjoint classes Periodic flows are such that each point of the space comes back to its initial position in a finite period of time.
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