Arbitrarily bad rates of convergence in Wasserstein metric

Question

Suppose $W_p(\mu_n,\mu)\to 0$ and $d(E(\mu_n),E(\mu))<r_n$. Here, $W_p$ is the $p$th-order Wasserstein distance (with respect to the metric $d$) and $\mu_n,\mu$ are probability measures on some metric space (say, $\mathbb{R}^n$, but this isn't crucial). Can anything be concluded regarding the convergence rate of $W_p(\mu_n,\mu)$? Or can this be arbitrarily bad? Does it help if $p=1$?

Convergence of the first moments is very weak, so my sense is that controlling the rate of convergence of the first moments won't tell us much about convergence of the underlying measures. I am curious if there is an explicit example to show how bad this can get. (Or, is my intuition off?)

Answer 1

Since you are talking about "[c]onvergence of the first moments", I will be assuming that by $E(\mu)$ you mean the mean of the probability measure $\mu$.

Anyhow, the convergence of the first moments does not help at all. For instance, let $\mu$ be the distribution of the constant-zero random variable (r.v.), that is, $\mu$ is the Dirac measure at $0$. For each natural $n$, let $\mu_n$ be the distribution of a r.v. taking each of the values $n$ and $-n$ with probability $1/2$. Then $d(E(\mu_n),E(\mu))=0$ for all natural $n$, whereas $W_p(\mu_n,\mu)=n\to\infty$ for each real $p>0$.

If you want to insist that $W_p(\mu_n,\mu)$ converge to $0$ but slower than any given sequence $t_n\downarrow0$, just replace the values $n$ and $-n$ of $X_n$ by $\sqrt{t_n}$ and $-\sqrt{t_n}$.