Canonical correlation for stochastic processes

A general notion of canonical correlation is developed that extends the classical multivariate concept to include function-valued random elements X and Y . The approach is based on the polar representation of a particular linear operator defined on reproducing kernel Hilbert spaces corresponding to the random functions X and Y . In this context, canonical correlations and variables are limits of finite-dimensional subproblems thereby providing a seamless transition between Hotelling’s original development and infinite-dimensional settings. Several infinite-dimensional treatments of canonical correlations that have been proposed for specific problems are shown to be special cases of this general formulation. We also examine our notion of canonical correlation from a large sample perspective and show that the asymptotic behavior of estimators can be tied to that of estimators from standard, finite-dimensional, multivariate analysis.