# Semi-additive functionals and cocycles in the context of self-similarity

Vladas Pipiras; Murad S. Taqqu

Discussiones Mathematicae Probability and Statistics (2010)

- Volume: 30, Issue: 2, page 149-177
- ISSN: 1509-9423

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topVladas Pipiras, and Murad S. Taqqu. "Semi-additive functionals and cocycles in the context of self-similarity." Discussiones Mathematicae Probability and Statistics 30.2 (2010): 149-177. <http://eudml.org/doc/277078>.

@article{VladasPipiras2010,

abstract = {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 properties. To prove its existence, we develop a general result about semi-additive functionals related to cocycles. The functional we identify, is helpful when solving for the kernel function generated by a flow. Its presence also sheds light on the previous results on the subject.},

author = {Vladas Pipiras, Murad S. Taqqu},

journal = {Discussiones Mathematicae Probability and Statistics},

keywords = {stable; self-similar processes with stationary increments; mixed moving averages; nonsingular flows; cocycles; semi-additive functionals; self-similar process with stationary increments},

language = {eng},

number = {2},

pages = {149-177},

title = {Semi-additive functionals and cocycles in the context of self-similarity},

url = {http://eudml.org/doc/277078},

volume = {30},

year = {2010},

}

TY - JOUR

AU - Vladas Pipiras

AU - Murad S. Taqqu

TI - Semi-additive functionals and cocycles in the context of self-similarity

JO - Discussiones Mathematicae Probability and Statistics

PY - 2010

VL - 30

IS - 2

SP - 149

EP - 177

AB - 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 properties. To prove its existence, we develop a general result about semi-additive functionals related to cocycles. The functional we identify, is helpful when solving for the kernel function generated by a flow. Its presence also sheds light on the previous results on the subject.

LA - eng

KW - stable; self-similar processes with stationary increments; mixed moving averages; nonsingular flows; cocycles; semi-additive functionals; self-similar process with stationary increments

UR - http://eudml.org/doc/277078

ER -

## References

top- [1] N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular Variation, Cambridge University Press 1987. Zbl0617.26001
- [2] I.P. Cornfeld, S.V. Fomin and Y.G. Sinai, Ergodic Theory, Springer-Verlag 1982.
- [3] C.D. Jr. Hardin, Isometries on subspaces of ${L}^{p}$, Indiana University Mathematics Journal 30 (1981), 449-465. Zbl0432.46026
- [4] S. Kolodyński and J. Rosiński, Group self-similar stable processes in ${\mathbb{R}}^{d}$, Journal of Theoretical Probability 16 (4) (2002), 855-876. Zbl1038.60038
- [5] I. Kubo, Quasi-flows, Nagoya Mathematical Journal 35 (1969), 1-30. Zbl0209.08903
- [6] I. Kubo, Quasi-flows II: Additive functionals and TQ-systems, Nagoya Mathematical Journal 40 (1970), 39-66. Zbl0213.07602
- [7] V. Pipiras and M.S. Taqqu, Decomposition of self-similar stable mixed moving averages, Probability Theory and Related Fields 123 (3)(2002 a), 412-452. Zbl1007.60026
- [8] V. Pipiras and M.S. Taqqu, The structure of self-similar stable mixed moving averages, The Annals of Probability 30 (2) (2002 b), 898-932. Zbl1016.60057
- [9] V. Pipiras and M.S. Taqqu, Stable stationary processes related to cyclic flows, The Annals of Probability 32 (3A) (2004), 2222-2260. Zbl1054.60056
- [10] Preprint. Available at http://www.stat.unc.edu/faculty/pipiras/preprints/articles.html.
- [11] J. Rosiński, On the structure of stationary stable processes, The Annals of Probability 23 (1995), 1163-1187. Zbl0836.60038
- [12] R.J. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, Boston 1984. Zbl0571.58015

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