# Structure of stationary stable processes

@inproceedings{Rosnski1995StructureOS, title={Structure of stationary stable processes}, author={Jan Ros{\'i}nski}, year={1995} }

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…

## 122 Citations

Stable stationary processes related to cyclic flows

- Mathematics
- 2004

We study stationary stable processes related to periodic and cyclic flows in the sense of Rosinski [Ann. Probab. 23 (1995) 1163–1187]. These processes are not ergodic. We provide their canonical…

Null flows, positive flows and the structure of stationary symmetric stable processes

- Mathematics
- 2004

This paper elucidates the connection between stationary symmetric α-stable processes with 0 < α < 2 and nonsingular flows on measure spaces by describing a new and unique decomposition of stationary…

Point processes associated with stationary stable processes

- Mathematics
- 2004

Point processes induced by stationary symmetric [alpha]-stable (S[alpha]S) processes can have diverse behavior. We distinguish two cases, depending on whether the stationary S[alpha]S process is…

On the mixing structure of stationary increment and self-similar symmetric \alpha-stable processes

- Mathematics
- 2012

Mixed moving average processes appear in the ergodic decomposition of stationary symmetric \alpha-stable (S\alpha S) processes. They correspond to the dissipative part of "deterministic" flows…

Decomposition of discrete time periodically correlated and multivariate stationary symmetric stable processes

- Mathematics
- 2005

The spectral structure of discrete time periodically correlated (as well as multivariate stationary) symmetric [alpha]-stable processes is identified by decomposing such a process uniquely in…

Decomposition of self-similar stable mixed moving averages

- Mathematics
- 2002

Abstract. Let α? (1,2) and Xα be a symmetric α-stable (S α S) process with stationary increments given by the mixed moving average
where is a standard Lebesgue space, is some measurable function…

On the structure and representations of max-stable processes

- MathematicsAdvances in Applied Probability
- 2010

We develop classification results for max-stable processes, based on their spectral representations. The structure of max-linear isometries and minimal spectral representations play important roles.…

Exceedance of power barriers for integrated continuous-time stationary ergodic stable processes

- MathematicsAdvances in Applied Probability
- 2009

We study the asymptotic behavior of the tail probability of integrated stable processes exceeding power barriers. In the first part of the paper the limiting behavior of the integrals of stable…

Decomposition of stationary $\alpha$-stable random fields

- Mathematics
- 2000

This work is concerned with the structural analysis of stationary a-stable random fields. Three distinct classes of such random fields are characterized and it is shown that every stationary a-stable…

Integral representations of periodic and cyclic fractional stable motions

- Mathematics
- 2004

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…