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We use $\mathcal{W}_2(\cdot, \cdot)$ to denote the quadratic Wasserstien distance as defined here. Now, let $X,Y = \mathcal{N}(0,1)$ be two standard normal random variables and for $ a \in[0,1]$ let $P_a$ be a discrete random variable, independent from $X,Y$, such that $$\mathbb{P}(P_a = 0) = 1-a \text{, } \mathbb{P}\left(P_a = \frac{1}{\sqrt{a}}\right) = a.$$ Consider the random variable $P_a\cdot X$, which is a centered mixture between a standard Gaussian, and a Gaussian with variance $\frac{1}{a}$. I'm intersted in estimating $$\mathcal{W}_2(P_a\cdot X, Y),$$ the quadratic Wasserstein distance between the mixture and a standard Gaussian.

Some easy bounds can be given by $$\mathcal{W}^2_2(P_a\cdot X, Y) \leq \mathbb{E}\left[(P_a - 1)^2\right]=a\left(\frac{1}{\sqrt{a}}-1\right)^2,$$ and $$\mathcal{W}^2_2(P_a\cdot X, Y) \geq \left(\sqrt{Var(P_a\cdot X)} - \sqrt{Var(Y)}\right)^2 = \left(\sqrt{2 - a} - 1\right)^2.$$

The two bounds leave a large gap between them, especially when $a$ is small, and I was wondering where the truth actually lies?

A more intricate question would be to ask for the Wasserstein distance between arbitrary mixtures, but that seems less tractable.

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