Abstract.
In this paper we derive the spectral and ergodic properties of a special class of homogeneous random fields, which includes an important family of evanescent random fields. Based on a derivation of the resolution of the identity for the operators generating the homogeneous field, and using the properties of measurable transformations, the spectral representation of both the field and its covariance sequence are derived. A necessary and sufficient condition for the existence of such representation is introduced. Using an analysis approach that employs the solution to the linear Diophantine equations, further characterization and modeling of the spectral properties of evanescent fields are provided by considering their spectral pseudo-density function, defined in this paper. The geometric properties of the spectral pseudo-density of the evanescent field are investigated. Finally, necessary and sufficient conditions for mean and strong ergodicity of the first and second order moments of these fields are derived. The analysis, initially carried out for complex valued random fields, is later extended to include the case of real valued fields.
References
Andrews, G.E.: Number Theory. W. B. Saunders Company. Philadelphia, 1971
Cheng, R., Houdré, C.: On the prediction of some L p random fields. J. Austral. Math. Soc. 67, 31–50 (1999)
Chiang, T.P.: The Prediction Theory of Stationary Random Fields III, Fourfold Wold Decompositions. J. Multivariate Anal. 37, 46–65 (1991)
Cuny, C.: Multidimensional Prediction Theory. Preprint
Doob, J.L.: Stochastic Processes. John Wiley & sons, 1953
Dunford, N., Schwartz, J.T.: Linear Operators Part I: General Theory. Wiley Classics Library, 1988
Francos, J.M., Nehorai, A.: Interference Mitigation in STAP using the Two-Dimensional Wold Decomposition Model. Accepted for publication, IEEE Trans. Signal Process
Francos, J.M., Meiri, A.Z., Porat, B.: A Wold-Like Decomposition of 2-D Discrete Homogenous Random Fields. Ann. Appl. Prob. 5, 248–260 (1995)
Francos, J.M., Narasimhan, A., Woods, J.W.: Maximum Likelihood Parameter Estimation of~ Textures Using a Wold Decomposition Based Model. IEEE Trans. Image Process. 4, 1655–1666 (1995)
Gaposhkin, V.F.: Criteria for the Strong Law of Large Numbers for some Classes of Second-Order Stationary Processes and Homogeneous Random Fields. Theory of Probability and its Applications XXII 2, 286–310 (1977)
Grenander, U., Rosenblatt, M.: Statistical Analysis of Stationary Time Series. New York: John Wiley & sons, 1957
Halmos, P.R.: Measure Theory. Graduate Texts in Mathematics, New York: Springer-Verlag, 1974
Halmos, P.R.: Lectures on Ergodic Theory. New York: Chelsea Publication company, 1956
Hanner, O.: Deterministic and non-deterministic stationary processes. Arkiv för Matematik 1, 161–177 (1950)
Helson, H., Lowdenslager, D.: Prediction Theory and Fourier Series in Several Variables. Acta Math. 99, 165–201 (1958)
Helson, H., Lowdenslager, D.: Prediction Theory and Fourier Series in Several Variables II. Acta Math. 106, 175–213 (1961)
Kallianpur, G., Miamee, A.G., Niemi, H.: On the Prediction Theory of Two-Parameter Stationary Random Fields. J. Multivariate Anal. 32, 120–149 (1990)
Korezlioglu, H., Loubaton, P.: Spectral Factorization of Wide Sense Stationary Processes on 2. J. Multivariate Anal. 19, 24–47 (1986)
Krengel, U.: Ergodic Theorems. de Gruyter Studies in Mathematics 6, 1985
Liu, F., Picard, R.W.: Periodicity, Directionality, and Randomness: Wold Features for Image Modeling and Retrieval. IEEE Trans. on Patt. Anal. Mach. Intell. 18, 722–733 (1996)
Picard, R.W.: A Society of Models for Video and Image Libraries. IBM Syst. J. 35, 292–312 (1996)
Riesz, F., Nagy, B.Sz.: Functional Analysis. Translated from the 2nd French edition by Leo F. Boron, Dover Publications Inc., 1990
Suciu, I.: On the Semi-group of Isometries. Studia Math. T. XXX, 101–110 (1968)
Zygmund, A.: Trigonometric Series. Cambridge: Cambridge University Press, 1977
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported in part by the EU 5th Framework IHP Program, MOUMIR Project, under Grant RTN-1999-0177.
Mathematics Subject Classification (2000):62M40, 62J05
Rights and permissions
About this article
Cite this article
Cohen, G., Francos, J. Spectral representation and asymptotic properties of certain deterministic fields with innovation components. Probab. Theory Relat. Fields 127, 277–304 (2003). https://doi.org/10.1007/s00440-003-0287-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-003-0287-x
Key words or phrases:
- Evanescent random fields
- Resolution of the identity
- Measurable transformations
- Spectral representation
- Linear Diophantine equations
- Ergodicity