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 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 are presented. The process is also related to multilinear forms on the target Hilbert space. Finally, a general structure result for involving the biorthogonal representation for the conditional expectation of with respect to the usual product -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