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 mu. Consider the empirical measures L-n = 1/n Sigma(n)(k=1) delta(Xk), 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 is an element of [1, +infinity)) if and only if integral(E) e(lambda dp(x0, x))d mu(x) < +infinity for all lambda > 0, and for some x(0) is an element of E. (C) 2009 Elsevier B.V. All rights reserved.