On measure-preserving transformations and doubly stationary symmetric stable processes

A. Gross; A. Weron

Studia Mathematica (1995)

  • Volume: 114, Issue: 3, page 275-287
  • ISSN: 0039-3223

Abstract

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In a 1987 paper, Cambanis, Hardin and Weron defined doubly stationary stable processes as those stable processes which have a spectral representation which is itself stationary, and they gave an example of a stationary symmetric stable process which they claimed was not doubly stationary. Here we show that their process actually had a moving average representation, and hence was doubly stationary. We also characterize doubly stationary processes in terms of measure-preserving regular set isomorphisms and the existence of σ-finite invariant measures. One consequence of the characterization is that all harmonizable symmetric stable processes are doubly stationary. Another consequence is that there exist stationary symmetric stable processes which are not doubly stationary.

How to cite

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Gross, A., and Weron, A.. "On measure-preserving transformations and doubly stationary symmetric stable processes." Studia Mathematica 114.3 (1995): 275-287. <http://eudml.org/doc/216192>.

@article{Gross1995,
abstract = {In a 1987 paper, Cambanis, Hardin and Weron defined doubly stationary stable processes as those stable processes which have a spectral representation which is itself stationary, and they gave an example of a stationary symmetric stable process which they claimed was not doubly stationary. Here we show that their process actually had a moving average representation, and hence was doubly stationary. We also characterize doubly stationary processes in terms of measure-preserving regular set isomorphisms and the existence of σ-finite invariant measures. One consequence of the characterization is that all harmonizable symmetric stable processes are doubly stationary. Another consequence is that there exist stationary symmetric stable processes which are not doubly stationary.},
author = {Gross, A., Weron, A.},
journal = {Studia Mathematica},
keywords = {invariant measures; nonsingular transformations; regular set isomorphisms; double stationarity},
language = {eng},
number = {3},
pages = {275-287},
title = {On measure-preserving transformations and doubly stationary symmetric stable processes},
url = {http://eudml.org/doc/216192},
volume = {114},
year = {1995},
}

TY - JOUR
AU - Gross, A.
AU - Weron, A.
TI - On measure-preserving transformations and doubly stationary symmetric stable processes
JO - Studia Mathematica
PY - 1995
VL - 114
IS - 3
SP - 275
EP - 287
AB - In a 1987 paper, Cambanis, Hardin and Weron defined doubly stationary stable processes as those stable processes which have a spectral representation which is itself stationary, and they gave an example of a stationary symmetric stable process which they claimed was not doubly stationary. Here we show that their process actually had a moving average representation, and hence was doubly stationary. We also characterize doubly stationary processes in terms of measure-preserving regular set isomorphisms and the existence of σ-finite invariant measures. One consequence of the characterization is that all harmonizable symmetric stable processes are doubly stationary. Another consequence is that there exist stationary symmetric stable processes which are not doubly stationary.
LA - eng
KW - invariant measures; nonsingular transformations; regular set isomorphisms; double stationarity
UR - http://eudml.org/doc/216192
ER -

References

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  1. P. Billingsley (1986), Probability and Measure, Wiley, New York. Zbl0649.60001
  2. S. Cambanis, C. D. Hardin, Jr. and A. Weron (1987), Ergodic properties of stationary stable processes, Stochastic Process. Appl. 24, 1-18. Zbl0612.60034
  3. A. Gross (1994), Some mixing conditions for stationary symmetric stable stochastic processes, ibid. 51, 277-285. Zbl0813.60039
  4. C. D. Hardin, Jr. (1981), Isometries on subspaces of L p , Indiana Univ. Math. J. 30, 449-465. Zbl0432.46026
  5. C. D. Hardin, Jr. (1982), On the spectral representation of symmetric stable processes, J. Multivariate Anal. 12, 385-401. Zbl0493.60046
  6. J. Lamperti (1958), On the isometries of certain function spaces, Pacific J. Math. 8, 459-466. Zbl0085.09702
  7. G. Maruyama (1970), Infinitely divisible processes, Probab. Theory Appl. 15, 3-23. Zbl0268.60036
  8. D. S. Ornstein (1960), On invariant measures, Bull. Amer. Math. Soc. 66, 297-300. Zbl0154.30502
  9. J. Rosinski (1994), On uniqueness of the spectral representation of stable processes, J. Theor. Probab. 7, 551-563. Zbl0805.60030

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