Identification of periodic and cyclic fractional stable motions

Vladas Pipiras; Murad S. Taqqu

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 4, page 612-637
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Pipiras, Vladas, and Taqqu, Murad S.. "Identification of periodic and cyclic fractional stable motions." Annales de l'I.H.P. Probabilités et statistiques 44.4 (2008): 612-637. <http://eudml.org/doc/77985>.

@article{Pipiras2008,
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.},
author = {Pipiras, Vladas, Taqqu, Murad S.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stable; self-similar processes with stationary increments; mixed moving averages; dissipative and conservative flows; periodic and cyclic flows; periodic and cyclic fractional stable motions; self-similar process; fractional stable motions},
language = {eng},
number = {4},
pages = {612-637},
publisher = {Gauthier-Villars},
title = {Identification of periodic and cyclic fractional stable motions},
url = {http://eudml.org/doc/77985},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Pipiras, Vladas
AU - Taqqu, Murad S.
TI - Identification of periodic and cyclic fractional stable motions
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 4
SP - 612
EP - 637
AB - 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.
LA - eng
KW - stable; self-similar processes with stationary increments; mixed moving averages; dissipative and conservative flows; periodic and cyclic flows; periodic and cyclic fractional stable motions; self-similar process; fractional stable motions
UR - http://eudml.org/doc/77985
ER -

References

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