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 $mu_n=(1/n),sum_i=1^ndelta_X_i$ the empirical measure. Then, $mu_n(B)overseta.s.longrightarrowmu(B)$ for all $Bsubset S$ and some (essentially unique) random probability measure $mu$. Denote by $mathcalL(Z)$ the probability distribution of any random variable $Z$. Under some assumptions on $mathcalL(mu)$, it is shown that eginequation* racanle hoigl[mathcalL(mu_n),,mathcalL(mu)igr]leracbnquad extandquad hoigl[mathcalL(mu_n),,mathcalL(a_n)igr]leraccn^u endequation* where $ ho$ is the bounded Lipschitz metric and $a_n(cdot)=Pigl(X_n+1incdotmid X_1,ldots,X_nigr)$ is the predictive measure. The constants $a,,b,,c>0$ and $uin (rac12, 1]$ depend on $mathcalL(mu)$ and card$,(S)$ only.