# zbMATH — the first resource for mathematics

Stable stationary processes related to cyclic flows. (English) Zbl 1054.60056
The authors study real- and complex-valued symmetric $$\alpha$$-stable stationary processes having an integral representation that is generated by periodic or cyclic flows. For general flows J. Rosiński [Ann. Probab. 23, No. 1, 464–473 (1995; Zbl 0831.60007) and ibid., No. 3, 1163–1187 (1995; Zbl 0836.60038)] obtained a unique decomposition in distribution of such processes into independent processes generated by a dissipative, respectively a conservative flow. Moreover, in the real-valued (resp. complex-valued) case, the conservative part can again be uniquely decomposed into a trivial (resp. harmonizable) process and an independent process having no trivial (resp. harmonizable) component.
The present paper refines the decomposition of Rosiński for special conservative flows, in which every space point comes back to its initial position after a finite (periodic flows) or positive and finite (cyclic flows) period in time. It is shown that the conservative part again can be decomposed into a trivial (resp. harmonizable) component and a cyclic component. Processes being representable in such a way are called stationary periodic processes, where the notion periodic refers to the flow. Among these, stationary cyclic processes fail to have a trivial (resp. harmonizable) part. In case of a minimal (nonredundant) representation, the authors prove that stationary periodic processes can only be generated by periodic flows, which fails for general representations. For general representations criteria others than the flow are provided to identify stationary periodic and cyclic processes among symmetric $$\alpha$$-stable stationary processes. For minimal representations, a further decomposition shows that the cyclic component still inhibits a nonperiodic conservative part which can be separated uniquely in distribution. Throughout the paper illustrative examples are given.

##### MSC:
 60G52 Stable stochastic processes 60G10 Stationary stochastic processes 37A50 Dynamical systems and their relations with probability theory and stochastic processes
Full Text:
##### References:
 [1] Arveson, W. (1976). An Invitation to $$C^\ast$$-Algebras . Springer, New York. · Zbl 0344.46123 [2] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation . Cambridge Univ. Press. · Zbl 0617.26001 [3] Cohn, D. L. (1972). Measurable choice of limit points and the existence of separable and measurable processes. Z. Wahrsch. Verw. Gebiete 22 161–165. · Zbl 0241.60025 [4] Gross, A. (1994). Some mixing conditions for stationary symmetric stable stochastic processes. Stochastic Process. Appl. 51 277–295. · Zbl 0813.60039 [5] Halmos, P. R. (1950). Measure Theory . Van Nostrand, New York. · Zbl 0040.16802 [6] Hardin, C. D., Jr. (1982). On the spectral representation of symmetric stable processes. J. Multivariate Anal. 12 385–401. · Zbl 0493.60046 [7] Krengel, U. (1985). Ergodic Theorems . de Gruyter, Berlin. · Zbl 0575.28009 [8] Kubo, I. (1969). Quasi-flows. Nagoya Math. J. 35 1–30. · Zbl 0209.08903 [9] Mackey, G. W. (1957). Borel structure in groups and their duals. Trans. Amer. Math. Soc. 85 134–165. · Zbl 0082.11201 [10] Petersen, K. (1983). Ergodic Theory . Cambridge Univ. Press. · Zbl 0507.28010 [11] Pipiras, V. and Taqqu, M. S. (2002a). Decomposition of self-similar stable mixed moving averages. Probab. Theory Related Fields 123 412–452. · Zbl 1007.60026 [12] Pipiras, V. and Taqqu, M. S. (2002b). The structure of self-similar stable mixed moving averages. Ann. Probab. 30 898–932. · Zbl 1016.60057 [13] Rosiński, J. (1995). On the structure of stationary stable processes. Ann. Probab. 23 1163–1187. JSTOR: · Zbl 0836.60038 [14] Rosiński, J. (1998a). Minimal integral representations of stable processes. · Zbl 1121.60032 [15] Rosiński, J. (1998b). Structure of stationary stable processes. In A Practical Guide to Heavy Tails : Statistical Techniques and Applications (R. Adler, R. Feldman and M. S. Taqqu, eds.) 461–472. Birkhäuser, Boston. · Zbl 0927.60051 [16] Rosiński, J. (2000). Decomposition of stationary $$\alpha$$-stable random fields. Ann. Probab. 28 1797–1813. · Zbl 1044.60039 [17] Rosiński, J. and Samorodnitsky, G. (1996). Classes of mixing stable processes. Bernoulli 2 365–377. · Zbl 0870.60032 [18] Rozanov, Y. A. (1967). Stationary Random Processes . Holden-Day, San Francisco. · Zbl 0152.16302 [19] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Processes : Stochastic Models with Infinite Variance . Chapman and Hall, New York. · Zbl 0925.60027 [20] Surgailis, D., Rosiński, J., Mandrekar, V. and Cambanis, S. (1998). On the mixing structure of stationary increment and self-similar $$\mathrmS \alpha\mathrmS$$ processes. [21] Walters, P. (1982). An Introduction to Ergodic Theory . Springer, New York. · Zbl 0475.28009 [22] Zimmer, R. J. (1984). Ergodic Theory and Semisimple Groups . Birkhäuser, Boston. · Zbl 0571.58015
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.