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

Let (Xn)n≥1(Xn)n≥1 be a sequence of i.i.d.r.v.’s with values in a Polish space (E,d) of law μμ. Consider the empirical measures Ln=1n∑k=1nδXk,n≥1. Our purpose is to generalize Sanov’s theorem about the large deviation principle of LnLn from the weak convergence topology to the stronger Wasserstein metric WpWp. We show that LnLn satisfies the large deviation principle in the Wasserstein metric WpWp (p∈[1,+∞)p∈[1,+∞)) if and only if ∫Eeλdp(x0,x)dμ(x)<+∞ for all λ>0λ>0, and for some x0∈Ex0∈E.