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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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