The Annals of Probability

On the Structure of Stationary Stable Processes

Jan Rosinski

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A connection between structural studies of stationary non-Gaussian stable processes and the ergodic theory of nonsingular flows is established and exploited. Using this connection, a unique decomposition of a stationary stable process into three independent stationary parts is obtained. It is shown that the dissipative part of a flow generates a mixed moving average part of a stationary stable process, while the identity part of a flow essentially gives the harmonizable part. The third part of a stationary process is determined by a conservative flow without fixed points and by a related cocycle.

Article information

Ann. Probab., Volume 23, Number 3 (1995), 1163-1187.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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

Stationary stable process spectral representation mixed moving average harmonizable process nonsingular flow Hopf decomposition cocycle


Rosinski, Jan. On the Structure of Stationary Stable Processes. Ann. Probab. 23 (1995), no. 3, 1163--1187. doi:10.1214/aop/1176988178.

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