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On the structure of stationary stable processes. (English) Zbl 0836.60038
The principal result of the paper is a unique in distribution decomposition of stationary symmetric non-Gaussian stable processes \(\{X_t, t \to T\}\) (with \(T = \mathbb{R}\) or \(T = \mathbb{Z})\) into three independent parts \(X =_d X^{(1)} + X^{(2)} + X^{(3)}\). \(\{X_t^{(1)}\}\) is a superposition of moving averages, \(\{X_t^{(2)}\}\) is a harmonizable process, and \(\{X_t^{ (3)}\}\) is a stationary stable process described by a conservative nonsingular flow without fixed points and by a related cocycle. The work is based on results of Hardin in minimal representations of stable processes. A spectral representation of a stationary stable process in terms of a nonsingular flow and a cocycle is established. The ergodic theory of nonsingular flows is exploited.

MSC:
60G10 Stationary stochastic processes
60G07 General theory of stochastic processes
60E07 Infinitely divisible distributions; stable distributions
60G57 Random measures
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