Rate of convergence of empirical measures for exchangeable sequences

Let $S$ be a finite set, $(X_n)$ an exchangeable sequence of $S$-valued random variables, and $\mu_n=(1/n)\,\sum_{i=1}^n\delta_{X_i}$ the empirical measure. Then, $\mu_n(B)\overset{a.s.}\longrightarrow\mu(B)$ for all $B\subset S$ and some (essentially unique) random probability measure $\mu$. Denote by $\mathcal{L}(Z)$ the probability distribution of any random variable $Z$. Under some assumptions on $\mathcal{L}(\mu)$, it is shown that \begin{equation*} \frac{a}{n}\le\rho\bigl[\mathcal{L}(\mu_n),\,\mathcal{L}(\mu)\bigr]\le\frac{b}{n}\quad\text{and}\quad\rho\bigl[\mathcal{L}(\mu_n),\,\mathcal{L}(a_n)\bigr]\le\frac{c}{n^u} \end{equation*} where $\rho$ is the bounded Lipschitz metric and $a_n(\cdot)=P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)$ is the predictive measure. The constants $a,\,b,\,c>0$ and $u\in (\frac{1}{2}, 1]$ depend on $\mathcal{L}(\mu)$ and card$\,(S)$ only.