Let $\mathcal{P}({\mathbb{R}^d})$ denote the set of Borel probability measures on $\mathbb{R}^d$, and let $\pi \in \mathcal{P} (\mathbb{R}^d)$. It is known that the sets $\{ \mu \in \mathcal{P}(\mathbb{R}^d) :\, KL( \mu \mid \pi ) \leq C\}$ are compact with respect to the weak convergence of probability measures, as well as w.r.t. setwise convergence. What about with respect to the 2-Wasserstein metric? (If one replaces $\mathcal{P}({\mathbb{R}^d})$ with $\mathcal{P}_2({\mathbb{R}^d})$.)
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$\begingroup$ What do you mean, $\pi\in \mathbb R^d$? Is $\pi$ a probability measure, or just a (posisbly unbounded, e.g. Lebesgue) nonnegative measure? $\endgroup$– leo monsaingeonCommented Jan 15, 2023 at 7:21
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$\begingroup$ Perhaps a relevant comment is that the $W_2$ distance is well-known to metrize "weak convergence + convergence of the 2nd moments"? $\endgroup$– leo monsaingeonCommented Jan 15, 2023 at 7:21
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$\begingroup$ Sorry, fixed typo regarding $\pi$. As for your second comment, I agree this is surely relevant, but do you have an idea of how to use this to resolve the issue? $\endgroup$– pseudocydoniaCommented Jan 15, 2023 at 17:20
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$\begingroup$ Well, this means that the only issue is a posisble escape of mass at infinity. $\endgroup$– leo monsaingeonCommented Jan 15, 2023 at 20:39
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$\begingroup$ Agreed. Do you have a proof that this is a non-issue, or know of a counter-example? $\endgroup$– pseudocydoniaCommented Jan 16, 2023 at 19:33
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