zbMATH — the first resource for mathematics

Identification of periodic and cyclic fractional stable motions. (English) Zbl 1179.60028
Authors’ abstract: We consider an important subclass of self-similar, non-Gaussian stable processes with stationary increments known as self-similar stable mixed moving averages. As previously shown by the authors, following the seminal approach of Jan Rosiński, these processes can be related to nonsingular flows through their minimal representations. Different types of flows give rise to different classes of self-similar mixed moving averages, and to corresponding general decompositions of these processes. Self-similar stable mixed moving averages related to dissipative flows have already been studied, as well as processes associated with identity flows which are the simplest type of conservative flows. The focus here is on self-similar stable mixed moving averages related to periodic and cyclic flows. Periodic flows are conservative flows such that each point in the space comes back to its initial position in finite time, either positive or null. The flow is cyclic if the return time is positive.
Self-similar mixed moving averages are called periodic, resp. cyclic, fractional stable motions if their minimal representations are generated by periodic, resp. cyclic, flows. In practice, however, minimal representations are not particularly easy to determine and, moreover, self-similar stable mixed moving averages are often defined by nonminimal representations. We therefore provide a way which is not based on flows, to detect whether these processes are periodic or cyclic even if their representations are nonminimal. These identification results lead naturally to a decomposition of self-similar stable mixed moving averages which includes the new classes of periodic and cyclic fractional stable motions, and hence is more refined than the one previously established.
60G52 Stable stochastic processes
60G18 Self-similar stochastic processes
28D15 General groups of measure-preserving transformations
37A50 Dynamical systems and their relations with probability theory and stochastic processes
Full Text: DOI EuDML arXiv
[1] J. Aaronson. An Introduction to Infinite Ergodic Theory . Amer. Math. Soc., Providence, RI, 1997. · Zbl 0882.28013
[2] P. Doukhan, G. Oppenheim and M. S. Taqqu, Eds. Long-Range Dependence: Theory and Applications . Birkhäuser, Boston, 2003.
[3] P. Embrechts and M. Maejima. Selfsimilar Processes . Princeton Univ. Press, 2002. · Zbl 1008.60003
[4] C. D. Hardin Jr. On the spectral representation of symmetric stable processes. J. Multivariate Anal. 12 (1982) 385-401. · Zbl 0493.60046
[5] U. Krengel. Ergodic Theorems . Walter de Gruyter, Berlin, 1985. · Zbl 0575.28009
[6] V. Pipiras and M. S. Taqqu. Decomposition of self-similar stable mixed moving averages. Probab. Theory Related Fields 123 (2002) 412-452. · Zbl 1007.60026
[7] V. Pipiras and M. S. Taqqu. The structure of self-similar stable mixed moving averages. Ann. Probab. 30 (2002) 898-932. · Zbl 1016.60057
[8] V. Pipiras and M. S. Taqqu. Dilated fractional stable motions. J. Theoret. Probab. 17 (2004) 51-84. · Zbl 1055.60041
[9] V. Pipiras and M. S. Taqqu. Stable stationary processes related to cyclic flows. Ann. Probab. 32 (2004) 2222-2260. · Zbl 1054.60056
[10] V. Pipiras and M. S. Taqqu. Integral representations for periodic and cyclic fractional stable motions. Electron. J. Probab. 12 (2007) 181-206. · Zbl 1130.60048
[11] G. Rangarajan and M. Ding, Eds. Processes with Long-Range Correlations: Theory and Applications. Lecture Notes in Phys. 621 . Springer, New York, 2003.
[12] J. Rosiński. On the structure of stationary stable processes. Ann. Probab. 23 (1995) 1163-1187. · Zbl 0836.60038
[13] J. Rosiński. Minimal integral representations of stable processes. Probab. Math. Statist. 26 (2006) 121-142. · Zbl 1121.60032
[14] G. Samorodnitsky and M. S. Taqqu. Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance . Chapman and Hall, New York, London, 1994. · Zbl 0925.60027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.