A note on ergodic symmetric stable processes. (English) Zbl 0758.60036
Summary: S. Cambanis, C. D. Hardin jun. and A. Weron [ibid. 24, 1-18 (1987; Zbl 0612.60034)] have characterized the ergodic symmetric stable processes in terms of their spectral representation. In this note we present another such characterization which contains only the one condition. We apply it to prove that notions of ergodicity and weak mixing coincide for this class of processes. Some examples of ergodic and non-ergodic processes are given.
MSC:
| 60G12 | General second-order stochastic processes |
| 60E07 | Infinitely divisible distributions; stable distributions |
| 60G10 | Stationary stochastic processes |
Keywords:
stationary processes; ergodic symmetric stable processes; spectral representation; ergodicity and weak mixingCitations:
Zbl 0612.60034References:
| [1] | Cambanis, S.; Hardin, C. D.; Weron, A., Ergodic properties of stationary stable processes, Stochastic Process. Appl., 24, 1-18 (1987) · Zbl 0612.60034 |
| [2] | Cornfeld, I. D.; Fomin, S. V.; Sinai, Ya. G., Ergodic Theory (1982), Springer: Springer Berlin · Zbl 0493.28007 |
| [3] | Hardin, C. D., On the spectral representation of symmetric stable processes, J. Multivariate Anal., 12, 384-401 (1982) · Zbl 0493.60046 |
| [4] | LePage, R., Multidimensional infinite divisible variables and processes, Part I: Stable case, (Lecture Nores in Math. (1989), Springer: Springer Berlin), 148-163, No. 1391 |
| [5] | Podgórski, K.; Weron, A., Characterization of ergodic stable processes via the dynamical functional: Stable processes and related topics, (Cambanis, S., Progress in Probability, 25 (1991), Birkhauser: Birkhauser Basel), 317-328 · Zbl 0723.60034 |
| [6] | G. Samorodnitsky and M. Taqqu, Stable processes, in preparation.; G. Samorodnitsky and M. Taqqu, Stable processes, in preparation. · Zbl 0925.60027 |
| [7] | Walters, P., An introduction to ergodic theory (1982), Springer: Springer Berlin · Zbl 0475.28009 |
| [8] | Weron, A., Stable processes and measures: A survey, (Lecture Notes in Math. (1984), Springer: Springer Berlin), 306-364, No. 1080 · Zbl 0548.60005 |
| [9] | Weron, A., Harmonizable stable processes on groups: Spectral, ergodic and interpolation properties, Z. Wahrsch. Verw. Gebiete, 68, 473-491 (1985) · Zbl 0537.60008 |
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