Asymptotic predictive inference with exchangeable data

Abstract

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 $a_{n}(\cdot)=P(X_{n+1}\in\cdot|\mathcal{G}_{n})$ be the empirical and the predictive measures. We focus on \begin{equation*}\Vert \mu_{n}-a_{n}\Vert =\mathop{\mathrm{sup}}_{B\in\mathcal{D}}\vert\mu_{n}(B)-a_{n}(B)\vert,\end{equation*} where $\mathcal{D}$ is a class of measurable sets. Conditions for $\Vert \mu_{n}-a_{n}\Vert \rightarrow0$, almost surely or in probability, are given. Also, to determine the rate of convergence, the asymptotic behavior of $r_{n}\Vert \mu_{n}-a_{n}\Vert $ 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.