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Anh, V. V.; Ruiz-Medina, M. D.; Angulo, J. M.

Covariance factorisation and abstract representation of generalised random fields. (English) Zbl 0968.60035

Bull. Aust. Math. Soc. 62, No. 2, 319-334 (2000).
Authors’ summary: This paper introduces a new concept of duality of generalized random fields using the geometric properties of Sobolev spaces of integer order. Under this duality condition, the covariance operators of a generalized random field and its dual can be factorized. The paper also defines a concept of generalized white noise relative to the geometries of the Sobolev spaces, and via the covariance factorization, obtains a representation of the generalized random field as a stochastic equation driven by a generalized white noise. This representation is unique except for isometric isomorphisms on the parameter space.
Reviewer: B.L.S.Prakasa Rao (New Delhi)

Cited in 8 Documents

MSC:

60G20 Generalized stochastic processes
60G60 Random fields

Keywords:

duality of generalized random fields; geometric properties of Sobolev spaces; covariance operators; generalized white noise
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References:

[1] Dym, Gaussian processes, function theory, and the inverse spectral problem (1976) · Zbl 0327.60029
[2] Gelfand, Generalized functions, Vol. 4 (1964)
[3] DOI: 10.1006/jmaa.1995.1162 · Zbl 0823.60037 · doi:10.1006/jmaa.1995.1162
[4] DOI: 10.1016/0304-4149(83)90052-2 · Zbl 0507.60031 · doi:10.1016/0304-4149(83)90052-2
[5] DOI: 10.1063/1.1699725 · doi:10.1063/1.1699725
[6] Wiener, The extrapolation, interpolation, and smoothing of stationary time series (1949) · Zbl 0036.09705
[7] Rozanov, Markov random fields (1982) · Zbl 0463.60047 · doi:10.1007/978-1-4613-8190-7
[8] Rozanov, Development in Statistics 2 pp 203– (1979)
[9] Ramm, Random fields estimation theory (1990) · Zbl 0712.47042
[10] Krein, Dokl. Akad. Nauk SSSR 93 pp 617– (1953)
[11] Krein, Dokl. Akad. Nauk SSSR 94 pp 987– (1954)
[12] Kolmogorov, Izu. Akad. Nauk SSSR 5 pp 3– (1941)
[13] Kolmogorov, Bjull. Moskov. Gosudarstv. Univ. Matematika 2 pp 1– (1941)
[14] Kolmogorov, C.R. Acad. Sci. Paris 208 pp 2043– (1939)
[15] DOI: 10.2307/1990716 · Zbl 0048.11202 · doi:10.2307/1990716
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