On the bracketing entropy condition and generalized empirical measures

TL;DR

This paper establishes Donsker and Glivenko--Cantelli theorems for generalized empirical measures under bracketing entropy conditions, and applies these results to posterior consistency and Bernstein--von Mises theorems for Dirichlet process priors on countable spaces.

Contribution

It extends classical empirical process results to generalized measures and derives new Bayesian asymptotic theorems under total variation topology.

Findings

Proves Donsker and Glivenko--Cantelli theorems for generalized measures.

Establishes posterior consistency and Bernstein--von Mises theorem for Dirichlet process on countable spaces.

Provides new insights into the Durst--Dudley--Borisov theorem.

Abstract

We prove a Donsker and a Glivenko--Cantelli theorem for sequences of random discrete measures generalizing empirical measures. Those two results hold under standard conditions upon bracketing numbers of the indexing class of functions. As a byproduct, we derive a posterior consistency and a Bernstein--von Mises theorem for the Dirichlet process prior, under the topology of total variation, when the observation space is countable. We also obtain new information about the Durst--Dudley--Borisov theorem