# Integral representations of periodic and cyclic fractional stable motions

@article{Pipiras2004IntegralRO, title={Integral representations of periodic and cyclic fractional stable motions}, author={Vladas Pipiras and Murad S.Taqqu}, journal={Electronic Journal of Probability}, year={2004}, volume={12}, pages={181-206} }

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 study. We focus on the structure of their integral representations and show that the periodic fractional stable motions have, in fact, a canonical representation. We study several examples and discuss questions of uniqueness, namely how to determine whether two given integral representations of periodic…

## 4 Citations

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Kernel functions of stable, self-similar mixed moving averages are known to be related to nonsingular flows. We identify and examine here a new functional occuring in this relation and study its…

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## References

SHOWING 1-10 OF 15 REFERENCES

Identification of periodic and cyclic fractional stable motions

- Mathematics
- 2004

Self-similar stable mixed moving average processes can be related to nonsingular flows through their minimal representations. Self-similar stable mixed moving averages related to dissipative flows…

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…

Structure of stationary stable processes

- Mathematics
- 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 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…

Dilated Fractional Stable Motions

- Mathematics
- 2004

Dilated fractional stable motions are stable, self-similar, stationary increments random processes which are associated with dissipative flows. Self-similarity implies that their finite-dimensional…

SEMI-ADDITIVE FUNCTIONALS AND COCYCLES IN THE CONTEXT OF SELF-SIMILARITY

- Mathematics
- 2004

Kernel functions of stable, self-similar mixed moving averages are known to be related to nonsingular flows. We identify and examine here a new functional occuring in this relation and study its…

MINIMAL INTEGRAL REPRESENTATIONS OF STABLE PROCESSES

- Mathematics
- 1998

Abstract: Minimal integral representations are defined for general st ochastic processes and completely characterized for stable processes ( symmetric and asymmetric). In the stable case, minimal…

The structure of self-similar stable mixed moving averages

- Mathematics
- 2002

Let fi2 (1;2) and Xfi be a symmetric fi-stable (SfiS) process with stationary increments given by the mixed moving average Xfi(t) = Z

Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance

- Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance
- 1994