Rate of convergence of predictive distributions for dependent data

TL;DR

This paper investigates the convergence rates of predictive distributions for dependent data, providing conditions under which empirical processes converge, with applications in Bayesian statistics.

Contribution

It establishes new conditions for the stable convergence of empirical processes of dependent data, including exchangeable and conditionally identically distributed sequences.

Findings

Conditions for stable convergence of empirical processes are derived.

Results show convergence in probability and almost sure convergence under certain conditions.

Applications to Bayesian statistical methods are discussed.

Abstract

This paper deals with empirical processes of the type \[C_n(B)=\sqrt{n}\{\mu_n(B)-P(X_{n+1}\in B\mid X_1,...,X_n)\},\] where $(X_n)$ is a sequence of random variables and $\mu_n=(1/n)\sum_{i=1}^n\delta_{X_i}$ the empirical measure. Conditions for $\sup_B|C_n(B)|$ to converge stably (in particular, in distribution) are given, where $B$ ranges over a suitable class of measurable sets. These conditions apply when $(X_n)$ is exchangeable or, more generally, conditionally identically distributed (in the sense of Berti et al. [Ann. Probab. 32 (2004) 2029--2052]). By such conditions, in some relevant situations, one obtains that $\sup_B|C_n(B)|\stackrel{P}{\to}0$ or even that $\sqrt{n}\sup_B|C_n(B)|$ converges a.s. Results of this type are useful in Bayesian statistics.