Pointwise limit laws for multivariate OT maps under realistic assumptions

Establish pointwise central limit theorems for estimators of multivariate optimal transport maps T₀ between distributions P and Q under more realistic modeling assumptions than those currently used, so that r_n (T̂(x₀) − T₀(x₀)) converges in distribution at fixed x₀ without requiring restrictive domain geometry or strong smoothness conditions.

Background

Existing pointwise limit laws for multivariate OT maps typically require smooth densities and a compact, regular domain such as the flat torus. These assumptions can be restrictive in practice, limiting the applicability of asymptotic inference tools for OT maps.

Developing pointwise CLTs under more realistic conditions would enable practical uncertainty quantification for multivariate OT maps across broader domains and with weaker regularity, aligning theory with applied needs in areas such as computational biology and physics where idealized assumptions rarely hold.