Asymptotic weak convergence holds for functionals of the empirical entropic optimal transport plan, supporting uniform confidence bands for colocalization curves via Hadamard differentiability and the delta method.
As it was pointed in Section 1, the mapψis well defined by existence and uniqueness of solution theorem for the entropic optimal transport problem (1) (see Nutz, 2021)
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Distributional Convergence of Empirical Entropic Optimal Transport and Statistical Applications
Asymptotic weak convergence holds for functionals of the empirical entropic optimal transport plan, supporting uniform confidence bands for colocalization curves via Hadamard differentiability and the delta method.