On the rate of convergence in de Finetti's representation theorem

TL;DR

This paper investigates the convergence rate of the empirical mean of exchangeable 0-1 variables to their de Finetti measure, providing explicit bounds and examples illustrating various convergence speeds.

Contribution

It extends existing results by establishing explicit $1/n$ bounds for smooth densities and explores the range of convergence rates in general cases.

Findings

Bounds of order 1/n for smooth densities in Wasserstein distance

Lower bound of 1/n and upper bound of 1/√n for general cases

Examples of exchangeable sequences with convergence rates between 1/n and 1/√n

Abstract

A consequence of de Finetti's representation theorem is that for every infinite sequence of exchangeable 0-1 random variables $(X_k)_{k\geq1}$, there exists a probability measure $\mu$ on the Borel sets of $[0,1]$ such that $\bar X_n = n^{-1} \sum_{i=1}^n X_i$ converges weakly to $\mu$. For a wide class of probability measures $\mu$ having smooth density on $(0,1)$, we give bounds of order $1/n$ with explicit constants for the Wasserstein distance between the law of $\bar X_n$ and $\mu$. This extends a recent result {by} Goldstein and Reinert \cite{goldstein2013stein} regarding the distance between the scaled number of white balls drawn in a P\'olya-Eggenberger urn and its limiting distribution. We prove also that, in the most general cases, the distance between the law of $\bar X_n$ and $\mu$ is bounded below by $1/n$ and above by $1/\sqrt{n}$ (up to some multiplicative constants). For every $\delta \in [1/2,1]$, we give an example of an exchangeable sequence such that this distance is of order $1/n^\delta$.