  We consider the problem of optimal transportation with general cost between a
empirical measure and a general target probability on R d , with d $\ge$ 1. We
extend results in [19] and prove asymptotic stability of both optimal transport
maps and potentials for a large class of costs in R d. We derive a central
limit theorem (CLT) towards a Gaussian distribution for the empirical
transportation cost under minimal assumptions, with a new proof based on the
Efron-Stein inequality and on the sequential compactness of the closed unit
ball in L 2 (P) for the weak topology. We provide also CLTs for empirical
Wassertsein distances in the special case of potential costs | $\bullet$ | p ,
p > 1.
