The following correction note was posted in September, 2013, and revised in December, 2014:

The statement of Theorem 1 contains an error, which was pointed out to me by Elisabeth Gassiat (who attributed it to Jonas Kahn). In particular, the Theorem 1 as stated only holds if $G_0$ is assumed to have exactly $k$ support points, but it does not hold if $G_0$ may have strictly less than $k$ support points.

There is a simple correction that suffices for the rest of the paper (specifically, only Theorem 5 on finite mixtures requires the result of Theorem 1; the remaining results are concerned with infinite mixtures instead): in the statement of Theorem 1, if we replace $G'$ by $G_0$, that is, we vary only $G$ while the other argument $G'$ is fixed to $G_0$, then the statement of the theorem continues to hold.

What this theorem establishes is a lower bound of $V(p_G,p_{G_0})$ in terms of Wasserstein distance $W_2(G,G_0)$, for a fixed $G_0$, in the overfitted setting. Moreover, the proof as published remains valid (by replacing every instance of $G'$ by $G_0$).

Thank you to Elisabeth and Jonas for pointing out the original mistake, and Nhat Ho for help with the correction.

Addendum (September, 2016): A generalization of Theorem 1 to the matrix-variate mixture model setting can be found in my recent EJS paper written with Nhat.