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Transactions of the American Mathematical Society

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Uncorrelatedness and orthogonality for vector-valued processes


Authors: Peter A. Loeb, Horst Osswald, Yeneng Sun and Zhixiang Zhang
Journal: Trans. Amer. Math. Soc. 356 (2004), 3209-3225
MSC (2000): Primary 03H05, 28E05, 47H60; Secondary 26E35
DOI: https://doi.org/10.1090/S0002-9947-03-03450-0
Published electronically: November 25, 2003
MathSciNet review: 2052947
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Abstract | References | Similar Articles | Additional Information

Abstract: For a square integrable vector-valued process $f$ on the Loeb product space, it is shown that vector orthogonality is almost equivalent to componentwise scalar orthogonality. Various characterizations of almost sure uncorrelatedness for $f$ are presented. The process $f$ is also related to multilinear forms on the target Hilbert space. Finally, a general structure result for $f$ involving the biorthogonal representation for the conditional expectation of $f$ with respect to the usual product $\sigma$-algebra is presented.


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Additional Information

Peter A. Loeb
Affiliation: Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, Illinois 61801
Email: loeb@math.uiuc.edu

Horst Osswald
Affiliation: Mathematisches Institut der LMU-München, Theresienstr.39, D-80333 München, Germany
Email: Horst.Osswald@mathematik.uni-muenchen.de

Yeneng Sun
Affiliation: Institute for Mathematical Sciences, National University of Singapore, 3 Prince George’s Park, Singapore 118402, Republic of Singapore, – and – Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore
Email: matsuny@nus.edu.sg

Zhixiang Zhang
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore, – and – School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China
Email: matzzx@nus.edu.sg

DOI: https://doi.org/10.1090/S0002-9947-03-03450-0
Keywords: Vector-valued processes, Loeb product space, Keisler's Fubini Theorem, orthogonality, uncorrelatedness, multilinear functional
Received by editor(s): March 11, 2003
Published electronically: November 25, 2003
Additional Notes: The authors are grateful for the support of the National University of Singapore during the initiation of this work
Article copyright: © Copyright 2003 American Mathematical Society

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