Uncorrelatedness and orthogonality for vector-valued processes

Abstract

For a square integrable vector-valued process f 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 f are presented. The process f f is also related to multilinear forms on the target Hilbert space. Finally, a general structure result for f f involving the biorthogonal representation for the conditional expectation of f f with respect to the usual product σ \sigma -algebra is presented.