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Uncorrelatedness and orthogonality for vector-valued processes |
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| Authors: | Peter A. Loeb Horst Osswald Yeneng Sun Zhixiang Zhang |
| Affiliation: | Department of Mathematics, University of Illinois, 1409 West Green St., Urbana, Illinois 61801 ; Mathematisches Institut der LMU-München, Theresienstr.39, D-80333 München, Germany ; 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 ; 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 |
| 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. |
| Keywords: | Vector-valued processes Loeb product space Keisler's Fubini Theorem orthogonality uncorrelatedness multilinear functional |
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