Asymptotic predictive inference with exchangeable data

Let $(X_n)$ be a sequence of random variables, adapted to a filtration $(\mathcal{G}_n)$, and let $\mu_n=(1/n)\,\sum_{i=1}^n\delta_{X_i}$ and $a_n(\cdot)=P(X_{n+1}\in\cdot\mid\mathcal{G}_n)$ be the empirical and the predictive measures. We focus on \begin{equation*} \norm{\mu_n-a_n}=\sup_{B\in\mathcal{D}}\,\abs{\mu_n(B)-a_n(B)} \end{equation*} where $\mathcal{D}$ is a class of measurable sets. Conditions for $\norm{\mu_n-a_n}\rightarrow 0$, almost surely or in probability, are given. Also, to determine the rate of convergence, the asymptotic behavior of $r_n\,\norm{\mu_n-a_n}$ is investigated for suitable constants $r_n$. Special attention is paid to $r_n=\sqrt{n}$ and $r_n=\sqrt{\frac{n}{\log\log n}}$. The sequence $(X_n)$ is exchangeable or, more generally, conditionally identically distributed.