Some asymptotic properties for functional canonical correlation analysis

•Prove upper bounds for the convergence rate of FCCA with two different losses.•Matching lower bounds is shown for the convergence rate.•A simulation is used to illustrate the difference of the two losses for tuning parameter choice.

We consider convergence rates of functional canonical correlation analysis (FCCA). There are already several studies on FCCA in the literature, which focused on its population properties as well as consistency. Our setup most closely resembles that of He et al. (2003). Under an assumption that controls the level of dependence (roughly that the dependence between the two functional objects is not too high), we derive convergence rates of the weight functions to their population counterpart. Both upper bound and lower bound are derived for the L2-normL2-norm and the prediction risk (also called Σ-norm) of the weight functions.