  We study statistical inference for the optimal transport (OT) map (also known
as the Brenier map) from a known absolutely continuous reference distribution
onto an unknown finitely discrete target distribution. We derive limit
distributions for the $L^p$-error with arbitrary $p \in [1,\infty)$ and for
linear functionals of the empirical OT map, together with their moment
convergence. The former has a non-Gaussian limit, whose explicit density is
derived, while the latter attains asymptotic normality. For both cases, we also
establish consistency of the nonparametric bootstrap. The derivation of our
limit theorems relies on new stability estimates of functionals of the OT map
with respect to the dual potential vector, which may be of independent
interest. We also discuss applications of our limit theorems to the
construction of confidence sets for the OT map and inference for a maximum tail
correlation. Finally, we show that, while the empirical OT map does not possess
nontrivial weak limits in the $L^2$ space, it satisfies a central limit theorem
in a dual H\"{o}lder space, and the Gaussian limit law attains the asymptotic
efficiency bound.
