Sanov's theorem in the Wasserstein distance: A necessary and sufficient condition

Let (Xn)n>=1 be a sequence of i.i.d.r.v.'s with values in a Polish space of law [mu]. Consider the empirical measures . Our purpose is to generalize Sanov's theorem about the large deviation principle of Ln from the weak convergence topology to the stronger Wasserstein metric Wp. We show that Ln satisfies the large deviation principle in the Wasserstein metric Wp (p[set membership, variant][1,+[infinity])) if and only if for all [lambda]>0, and for some x0[set membership, variant]E.