Wasserstein-type concentration inequalities for empirical measures on polish spaces

Question

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.

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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.

Answer 1

Yes, there are various results available in more general settings. The typical route would be to combine an upper bound on the expected distance between the law and the empirical measure (like Theorem 1.1 in Boissard and Le Gouic) with a concentration estimate around this expectation (like Proposition 20 in Weed and Bach). In particular, both of these results work on a general metric space which is totally bounded.

I do not know of a result like the one you ask for that works in an arbitrary Polish space; I expect one would need to impose further regularity assumptions on the law of $X_1$.