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Some properties of canonical correlations and variates in infinite dimensions. (English) Zbl 1141.62046
Summary: The notion of functional canonical correlation as a maximum of correlations of linear functionals is explored. It is shown that the population functional canonical correlation is in general well defined, but that it is a supremum rather than a maximum, so that a pair of canonical variates may not exist in the spaces considered. Also the relation with the maximum eigenvalue of an associated pair of operators and the corresponding eigenvectors is not in general valid. When the inverses of the operators involved are regularized, however, all of the above properties are restored. Relations between the actual population quantities and their regularized versions are also established. The sample functional canonical correlations can be regularized in a similar way, and consistency is shown at a fixed level of the regularization parameter.

MSC:
62H20 Measures of association (correlation, canonical correlation, etc.)
62M99 Inference from stochastic processes
47N30 Applications of operator theory in probability theory and statistics
46N30 Applications of functional analysis in probability theory and statistics
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