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

Let ( X n ) n ≥ 1 be a sequence of i.i.d.r.v.’s with values in a Polish space ( E , d ) of law μ . Consider the empirical measures L n = 1 n ∑ k = 1 n δ X k , n ≥ 1 . Our purpose is to generalize Sanov’s theorem about the large deviation principle of L n from the weak convergence topology to the stronger Wasserstein metric W p . We show that L n satisfies the large deviation principle in the Wasserstein metric W p ( p ∈ [ 1 , + ∞ ) ) if and only if ∫ E e λ d p ( x 0 , x ) d μ ( x ) + ∞ for all λ > 0 , and for some x 0 ∈ E .