For a family of probability measures sharing the same form of distribution function $F(x; p)$ with different parameters (i.e., $p$'s), if the parameter falls in a compact subset of real line, can we say these probability measures constitute a compact subset in Wasserstein space?


For example, considering exponential distributions $\text{Exp}(\lambda)$ and $\lambda \in [a, b] \subset \mathbb{R}$, intuitively, it seems these exponential distributions should be in a compact subset in the sense of Wasserstein metric. If so, I am just wondering whether there is any rigorous proof for it.



Does the compactness of parameter of distribution function imply the compactness of the distribution (or probability measure) in Wasserstein space?
Since the continuous image of a compact set is compact, it suffices to determine whether the mapping $p \to F(x,p)$ is continuous. This is the case for most natural parametrized families of distributions, but needs to be verified in each case. For instance, for the exponential family, the transformation $x \mapsto(1+\epsilon) x$ takes the Exp$(\lambda)$ distribution $\mu$ (which has mean $1/\lambda$) to the Exp$(\frac{\lambda}{1+\epsilon})$ distribution $\nu$. The Wasserstein distance can be bounded using this coupling:
$$W\_1(\mu,\nu) \le \int |\epsilon x| \, d\mu(x)=|\epsilon/\lambda| \,.$$
This suffices to establish continuity of the mapping $\lambda \mapsto$ Exp$(\lambda)$.


