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Harmonizable stable fields: Regularity and Wold-type decompositions


Author: David A. Redett
Journal: Proc. Amer. Math. Soc. 146 (2018), 831-844
MSC (2010): Primary 60G60, 60G52
DOI: https://doi.org/10.1090/proc/13812
Published electronically: October 12, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: In this article, we examine the structure of harmonizable stable fields. We start by examining horizontal and vertical regularity. We find equivalent conditions for horizontal and vertical regularity in terms of the harmonizable stable field's spectral measure. We then give a Wold-type decomposition in this setting. After that, we consider strong regularity. Here too, we give equivalent conditions for strong regularity in terms of the field's spectral measure. In addition, we show that strong regularity is equivalent to the field's ability to be represented by a moving average random field. We finish this article with a four-fold Wold-type decomposition.


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  • [1] S. Cambanis and A. G. Miamee, On prediction of harmonizable stable processes, Sankhyā Ser. A 51 (1989), no. 3, 269-294. MR 1175606
  • [2] Stamatis Cambanis and A. Reza Soltani, Prediction of stable processes: spectral and moving average representations, Z. Wahrsch. Verw. Gebiete 66 (1984), no. 4, 593-612. MR 753815, https://doi.org/10.1007/BF00531892
  • [3] Raymond Cheng, Weakly and strongly outer functions on the bidisc, Michigan Math. J. 39 (1992), no. 1, 99-109. MR 1137892, https://doi.org/10.1307/mmj/1029004458
  • [4] Clyde D. Hardin Jr., On the spectral representation of symmetric stable processes, J. Multivariate Anal. 12 (1982), no. 3, 385-401. MR 666013, https://doi.org/10.1016/0047-259X(82)90073-2
  • [5] Yuzo Hosoya, Harmonizable stable processes, Z. Wahrsch. Verw. Gebiete 60 (1982), no. 4, 517-533. MR 665743, https://doi.org/10.1007/BF00535714
  • [6] Robert C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265-292. MR 0021241, https://doi.org/10.2307/1990220
  • [7] G. Kallianpur and V. Mandrekar, Nondeterministic random fields and Wold and Halmos decompositions for commuting isometries, Prediction theory and harmonic analysis, North-Holland, Amsterdam, 1983, pp. 165-190. MR 708524
  • [8] G. Kallianpur, A. G. Miamee, and H. Niemi, On the prediction theory of two-parameter stationary random fields, J. Multivariate Anal. 32 (1990), no. 1, 120-149. MR 1035612, https://doi.org/10.1016/0047-259X(90)90076-T
  • [9] A. Makagon and V. Mandrekar, The spectral representation of stable processes: harmonizability and regularity, Probab. Theory Related Fields 85 (1990), no. 1, 1-11. MR 1044294, https://doi.org/10.1007/BF01377623
  • [10] V. Mandrekar and D. A. Redett, Weakly Stationary Random Fields, Invariant Subspaces and Applications, (in progress).
  • [11] Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
  • [12] K. Urbanik, Random measures and harmonizable sequences, Studia Math. 31 (1968), 61-88. MR 0246340

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Additional Information

David A. Redett
Affiliation: Department of Mathematical Sciences, IPFW, 2101 E. Coliseum Boulevard, Fort Wayne, Indiana 46805-1499
Address at time of publication: Department of Mathematics, Terra State Community College, 2830 Napoleon Road, Fremont, Ohio 43420
Email: redettd@gmail.com

DOI: https://doi.org/10.1090/proc/13812
Keywords: Harmonizable stable fields, regularity, Wold-type decompositions
Received by editor(s): July 2, 2016
Received by editor(s) in revised form: March 17, 2017
Published electronically: October 12, 2017
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2017 American Mathematical Society

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