Connection between Wassertein-2 metric and difference in variance

Question

Given two probability densities $\mu\in\mathcal P(\mathbb R^d)$ and $\nu\in\mathcal P(\mathbb R^d)$, we define their Wasserstein-$p$ metric as $$ W_p^p(\mu, \nu)=\inf_{\gamma\in \Gamma(\mu, \nu)}\int_{\mathbb R^d\times \mathbb R^d}|x-y|^pd\gamma(x, y), $$ where $\Gamma(\mu, \nu)$ represents the set of couplings between $\mu$ and $\nu$.

For $p=1$, the Kantorovich-Rubenstein duality allows us to bound the difference in first moments of $\mu$ and $\nu$ by the $W_1$ distance between them. Specifically, $$ |\mu(\text{Id})-\nu(\text{Id})|\le W_1(\mu, \nu). $$ Is there an analogous result, bounding the first and second moments of $\mu$ and $\nu$ (expectation and variance) by the $W_2$-distance between them?

Answer 1

Let $\mu_k$ and $\nu_k$ denote the $k$th moments of $\mu$ and $\nu$ respectively.

Write $$W_p(\mu,\nu)=\inf\{\|X-Y\|_p\colon X\sim\mu,Y\sim\nu\},$$ where $\|Z\|_p:=(E|Z|^p)^{1/p}$.

$\newcommand\si\sigma$Then for any random variables $X$ and $Y$ such that $X\sim\mu$ and $Y\sim\nu$ we have $$|\mu_2^{1/2}-\nu_2^{1/2}|=|\|X\|_2-\|Y\|_2|\le\|X-Y\|_2$$ and $$|\mu_1-\nu_1|=|EX-EY|\le\|X-Y\|_1\le\|X-Y\|_2.$$

So, $$|\mu_2^{1/2}-\nu_2^{1/2}|\le W_2(\mu,\nu) \tag{1}\label{1}$$ and $$|\mu_1-\nu_1|\le W_2(\mu,\nu).$$

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Concerning \eqref{1}, note that $|\mu_2-\nu_2|$ cannot be bounded in terms of $W_2(\mu,\nu)$. Indeed, if e.g. $\mu$ and $\nu$ are the centered normal distributions over $\Bbb R$ with respective standard deviations $\si+1$ and $\si$, then, by Proposition 7, $W_2(\mu,\nu)=1$, whereas $|\mu_2-\nu_2|=2\si+1\to\infty$ as $\si\to\infty$.