# Wasserstein-type concentration inequalities for empirical measures on polish spaces

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.

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 $$X_1$$.