Asymptotics of Predictive Distributions

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 $a_n(\cdot )=P(X_{n+1}\in \cdot \mid \mathcal {G}_n)$ be the empirical and the predictive measures. We focus on $||\mu _n-a_n||=\sup _{B\in \mathcal {D}}\,|\mu _n(B)-a_n(B)|$, where $\mathcal {D}$ is a class of measurable sets. Conditions for $||\mu _n-a_n||\rightarrow 0$, almost surely or in probability, are given. Also, to determine the rate of convergence, the asymptotic behavior of $r_n\,||\mu _n-a_n||$ is investigated for suitable constants $r_n$. Special attention is paid to $r_n=\sqrt{n}$. The sequence $(X_n)$ is exchangeable or, more generally, conditionally identically distributed.