Spectral representation and extrapolation of stationary random processes on linear spaces

• Autorzy: Klotz L. , Riedel M.
• Języki publikacji: EN

Abstrakty

EN

The paper deals with continuous Banach-space-valued stationary random processes on linear spaces. From von Waldenfels’ [13] integral representation of positive definite functions on a linear space L we derive an analogue of Stone’s theorem for a group of unitary operators over L. It is used to obtain spectral representations of a general Banach-space-valued stationary random process over L and its covariance function. For the special class of Hilbert-Schmidt operator-valued stationary processes the explicit form of Kolmogorov’s isomorphism theorem between temporal space and spectral space is established and with its aid there are studied some prediction problems. Our prediction results are similar to those proved in [5] for multivariate stationary processes on groups.

Słowa kluczowe

EN

Banach space; Hilbert-Schmidt operator; Kolmogorov’s isomorphism

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2001
• Tom: Vol. 21, Fasc. 1
• Strony: 179--197
• Opis fizyczny: Biblogr. 13 poz.

Twórcy

autor

Klotz L.

• Fakultät für Mathematik und Informatik Universität Leipzig 04109 Leipzig, Germany

autor

Riedel M.

• Fakultät für Mathematik und Informatik Universität Leipzig 04109 Leipzig, Germany

Bibliografia

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• [3] J. Dixmier, Von Neumann Algebras, North-Holland, Amsterdam 1981.
• [4] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Pure Appl. Math., Vol. 7, Interscience Publishers, 2nd edition, New York 1964.
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• [6] I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Non-selfadjoint Operators in Hilbert Space (in Russian), Nauka, Moscow 1965.
• [7] P. Hennequin and A. Tortrat, Probability Theory and Its Applications (in Russian), Nauka, Moscow 1974.
• [8] L. Klotz, Some Banach spaces of measurable operator-valued functions, Probab. Math. Statist. 12 (1991), pp. 85-97.
• [9] V. Mandrekar and H. Salehi, The square integrability of operator-valued functions with respect to a non-negative operator-valued measure and the Kolmogorov isomorphism theorem, Indiana Univ. Math. J. 20 (1970-1971), pp. 545-563.
• [10] R. Payen, Fonctions aléatoires du second ordre á valeurs dans une espace de Hilbert, Ann, Inst. H. Poincare Probab. Statist. 3 (1967), pp. 323-396.
• [11] H. Salehi, Stone's theorem for a group of unitary operators over a Hilbert space, Proc. Amer. Math. Soc. 31 (1972), pp. 480-484.
• [12] F. Schmidt, Spektraldarstellung und Extrapolation einer Klasse von stationären stochastischen Prozessen, Math. Nachr. 47 (1970), pp. 101-119.
• [13] W. von Waldenfels, Positiv definite Funktionen auf einem unendlich dimensionalen Vektorraum, Studia Math. 30 (1968), pp. 153-162.