The hierarchy of chaotic properties of symmetric infinitely divisible stationary processes is studied in the language of their stochastic representation. The structure of the Musielak-Orlicz space in this representation is exploited here.
@article{bwmeta1.element.bwnjournal-article-smv115i2p109bwm,
author = {S. Cambanis and K. Podg\'orski and A. Weron},
title = {Chaotic behavior of infinitely divisible processes},
journal = {Studia Mathematica},
volume = {113},
year = {1995},
pages = {109-127},
zbl = {0835.60008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv115i2p109bwm}
}
Cambanis, S.; Podgórski, K.; Weron, A. Chaotic behavior of infinitely divisible processes. Studia Mathematica, Tome 113 (1995) pp. 109-127. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv115i2p109bwm/
[00000] R. J. Adler, S. Cambanis and G. Samorodnitsky (1990), On stable Markov processes, Stochastic Process. Appl. 34, 1-17. | Zbl 0692.60054
[00001] R. M. Blumenthal (1957), An extended Markov property, Trans. Amer. Math. Soc. 85, 52-72. | Zbl 0084.13602
[00002] S. Cambanis, C. D. Hardin and A. Weron (1987), Ergodic properties of stationary stable processes, Stochastic Process. Appl. 24, 1-18. | Zbl 0612.60034
[00003] S. Cambanis and A. Ławniczak (1989), Ergodicity and mixing of infinitely divisible processes, unpublished preprint.
[00004] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai (1982), Ergodic Theory, Springer, Berlin. | Zbl 0493.28007
[00005] S. V. Fomin (1950), Normal dynamical systems, Ukrain. Mat. Zh. 2, 25-47 (in Russian).
[00006] U. Grenander (1950), Stochastic processes and statistical inference, Ark. Mat. 1, 195-277. | Zbl 0041.45807
[00007] A. Gross (1994), Some mixing conditions for stationary symmetric stable stochastic processes, Stochastic Process. Appl. 51, 277-285. | Zbl 0813.60039
[00008] A. Gross and J. B. Robertson (1993), Ergodic properties of random measures on stationary sequences of sets, ibid. 46, 249-265. | Zbl 0786.60044
[00009] M. Hernández and C. Houdré (1993), Disjointness results for some classes of stable processes, Studia Math. 105 235-252. | Zbl 0810.60044
[00010] P. Kokoszka and K. Podgórski (1992), Ergodicity and weak mixing of semistable processes, Probab. Math. Statist. 13, 239-244. | Zbl 0781.60029
[00011] A. Lasota and M. C. Mackey (1994), Chaos, Fractals and Noise. Stochastic Aspects of Dynamics, Springer, New York. | Zbl 0784.58005
[00012] V. P. Leonov (1960), The use of the characteristic functional and semiinvariants in the theory of stationary processes, Dokl. Akad. Nauk SSSR 133, 523-526 (in Russian).
[00013] G. Maruyama (1949), The harmonic analysis of stationary stochastic processes, Mem. Fac. Sci. Kyusyu Ser. Mat. IV 1, 49-106. | Zbl 0045.40602
[00014] G. Maruyama (1970), Infinitely divisible processes, Probab. Theory Appl. 15, 3-23. | Zbl 0268.60036
[00015] J. Musielak (1983), Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, New York. D. Newton (1968), On a principal factor system of a normal dynamical system, J. London Math. Soc. 43, 275-279. | Zbl 0169.20601
[00016] K. Podgórski (1992), A note on ergodic symmetric stable processes, Stochastic Process. Appl. 43, 355-362. | Zbl 0758.60036
[00017] K. Podgórski and A. Weron (1991), Characterization of ergodic stable processes via the dynamical functional, in: Stable Processes and Related Topics, S. Cambanis et al. (eds.), Birkhäuser, Boston, 317-328. | Zbl 0723.60034
[00018] B. S. Rajput and J. Rosiński (1989), Spectral representations of infinitely divisible processes, Probab. Theory Related Fields 82, 451-487. | Zbl 0659.60078
[00019] V. A. Rokhlin (1964), Exact endomorphisms of Lebesgue space, Amer. Math. Soc. Transl. (2) 39, 1-36. | Zbl 0154.15703
[00020] P. Walters (1982), An Introduction to Ergodic Theory, Springer, Berlin. | Zbl 0475.28009
[00021] A. Weron (1985), Harmonizable stable processes on groups: spectral, ergodic and interpolation properties, Z. Wahrsch. Verw. Gebiete 68, 473-491. | Zbl 0537.60008