For random variables $X_1,\dots,X_n$ with common mean $\mathbb{E}[X_i]=\mu$ and common bounds $a\leq X_i\leq b$, we have the very useful Hoeffding's inequality: $$\mathbb{P}\left(\left|\mu -\frac1n\sum_iX_i \right|\geq t\right)\leq 2\exp\left(-\frac{2nt^2}{(b-a)^2}\right).$$
I am interested in the following case: let $\nu_1,\dots,\nu_n$ be random measures (that is $\nu_i:\Omega \to \mathscr{P}(\mathbb{R})$), such that $\mathbb{E}[\nu_i]=\nu$ (where $\mathbb{E}(\nu_i):= \sum_{\omega\in \Omega} \mathbb{P}(\omega)\nu_i(\omega)$). Moreover, we have a similar boundedness assumption, $\text{supp}(\nu_i)\subseteq[a,b]$. Is there a similar concentration inequality such as Hoeffding's that can be applied in this case?
In particular, I am interested in whether one can say $$\mathbb{P}\left(\mathcal{W}\left(\nu,\frac1n\sum_i\nu_i \right)\geq t\right)\leq 2\exp\left(-\frac{2nt^2}{(b-a)^2}\right),$$ where $\mathcal{W}$ is the Wasserstein distance.
