Harmonizable stable fields: Regularity and Wold-type decompositions
HTML articles powered by AMS MathViewer
- by David A. Redett PDF
- Proc. Amer. Math. Soc. 146 (2018), 831-844 Request permission
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.References
- S. Cambanis and A. G. Miamee, On prediction of harmonizable stable processes, Sankhyā Ser. A 51 (1989), no. 3, 269–294. MR 1175606
- 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, DOI 10.1007/BF00531892
- Raymond Cheng, Weakly and strongly outer functions on the bidisc, Michigan Math. J. 39 (1992), no. 1, 99–109. MR 1137892, DOI 10.1307/mmj/1029004458
- Clyde D. Hardin Jr., On the spectral representation of symmetric stable processes, J. Multivariate Anal. 12 (1982), no. 3, 385–401. MR 666013, DOI 10.1016/0047-259X(82)90073-2
- Yuzo Hosoya, Harmonizable stable processes, Z. Wahrsch. Verw. Gebiete 60 (1982), no. 4, 517–533. MR 665743, DOI 10.1007/BF00535714
- Robert C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc. 61 (1947), 265–292. MR 21241, DOI 10.1090/S0002-9947-1947-0021241-4
- 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
- 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, DOI 10.1016/0047-259X(90)90076-T
- 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, DOI 10.1007/BF01377623
- V. Mandrekar and D. A. Redett, Weakly Stationary Random Fields, Invariant Subspaces and Applications, (in progress).
- Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
- K. Urbanik, Random measures and harmonizable sequences, Studia Math. 31 (1968), 61–88. MR 246340, DOI 10.4064/sm-31-1-61-88
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
- MR Author ID: 751935
- Email: redettd@gmail.com
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 831-844
- MSC (2010): Primary 60G60, 60G52
- DOI: https://doi.org/10.1090/proc/13812
- MathSciNet review: 3731715