  We develop a framework of canonical correlation analysis for
distribution-valued functional data within the geometry of Wasserstein spaces.
Specifically, we formulate an intrinsic concept of correlation between random
distributions, propose estimation methods based on functional principal
component analysis (FPCA) and Tikhonov regularization, respectively, for the
correlation and its corresponding weight functions, and establish the minimax
convergence rates of the estimators. The key idea is to extend the framework of
tensor Hilbert spaces to distribution-valued functional data to overcome the
challenging issue raised by nonlinearity of Wasserstein spaces. The
finite-sample performance of the proposed estimators is illustrated via
simulation studies, and the practical merit is demonstrated via a study on the
association of distributions of brain activities between two brain regions.
