Let $(\mathcal{X},d)$ be a Polish (metric) space and let $\{X_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. $\mathcal{X}$-valued random elements defined on a common complete (standard) probability space $(\Omega,\mathcal{F},\mathbb{P})$ with $\mathbb{E}[d(X_1,x)^p]<\infty$ for some $p>1$ and some $x\in \mathcal{X}$.

Are the known concentration inequalities of the form: $$ \mathbb{P}\left( \mathcal{W}_1\left(\frac1{n}\sum_{k=1}^n \delta_{X_k},Law(X_1)\right)>\delta \right) \leq I(n,\delta), $$ for some ``well-behaved'' function $I:\mathbb{N}\times (0,\infty)\rightarrow \infty)$ which is:

- Monotone decreasing in its both its arguments and converges to $0$,
- Upper semi-continuous in its second argument.

**Additional Piece of Information:** I'm not interested in *best/sharp* rates, I only really case about some quantitative rates of the above form.

**What I've seen so far:**

I only know of this 2015 PTRF article which describes sharp rates in the case where $(\mathcal{X},d)$ is a finite-dimensional Euclidean space.