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Covariance factorisation and abstract representation of generalised random fields

Published online by Cambridge University Press:  17 April 2009

V. V. Anh ,
M. D. Ruiz-Medina  and
J. M. Angulo
V. V. Anh
Affiliation:
Centre In Statistical Science and Industrial Mathematics, Queensland University of Technology, GPO Box 2434, Brisbane Qld. 4001, Australia e-mail: [email protected]
M. D. Ruiz-Medina
Affiliation:
Centre In Statistical Science and Industrial Mathematics, Queensland University of Technology, GPO Box 2434, Brisbane Qld. 4001, Australia e-mail: [email protected]
J. M. Angulo
Affiliation:
Department of Statistics & Operations Research, University of Granada, Campus Fuente Nueva s/n, E-18071 Granada, Spain e-mail: [email protected]@goliat.ugr.es
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Abstract

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This paper introduces a new concept of duality of generalised random fields using the geometric properties of Sobolev spaces of integer order. Under this duality condition, the covariance operators of a generalised random field and its dual can be factorised. The paper also defines a concept of generalised white noise relative to the geometries of the Sobolev spaces, and via the covariance factorisation, obtains a representation of the generalised random field as a stochastic equation driven by a generalised white noise. This representation is unique except for isometric isomorphisms on the parameter space.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 62 , Issue 2 , October 2000 , pp. 319 - 334
Copyright
Copyright © Australian Mathematical Society 2000

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