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The spectral representation of stable processes: Harmonizability and regularity. (English) Zbl 0673.60041

We show that symmetric \(\alpha\)-stable moving average processes are not harmonizable. However, we show that a concept of generalized spectrum holds for all \(L_ p\)-bounded processes, \(0<p\leq 2\). In case \(p=2\), the generalized spectrum is a measure and the classical representation follows. For strongly harmonizable symmetric \(\alpha\)-stable processes we derive necessary and sufficient conditions for the regularity and the singularity for \(0<\alpha \leq 2\) using known results on the invariant subspaces. We also get a Cramér-Wold decomposition for the case \(0<\alpha \leq 2\).
Reviewer: A.Makagon

MSC:

60G12 General second-order stochastic processes
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