De Finetti's theorem: rate of convergence in Kolmogorov distance

TL;DR

This paper studies the rate at which the average of exchangeable Bernoulli variables converges to its limit distribution in Kolmogorov distance, providing new bounds and conditions for optimal convergence rates.

Contribution

It offers a quantitative analysis of de Finetti's theorem, establishing convergence rates and weaker regularity conditions for the distribution of the limiting variable.

Findings

Any rate of 1/n^α for α in (0,1] can be achieved.

A sufficient condition on Y's distribution for the optimal 1/n convergence rate.

Improves existing results by studying a stronger metric and weakening regularity assumptions.

Abstract

This paper provides a quantitative version of de Finetti law of large numbers. Given an infinite sequence $\{X_n\}_{n \geq 1}$ of exchangeable Bernoulli variables, it is well-known that $\frac{1}{n} \sum_{i = 1}^n X_i \stackrel{a.s.}{\longrightarrow} Y$, for a suitable random variable $Y$ taking values in $[0,1]$. Here, we consider the rate of convergence in law of $\frac{1}{n} \sum_{i = 1}^n X_i$ towards $Y$, with respect to the Kolmogorov distance. After showing that any rate of the type of $1/n^{\alpha}$ can be obtained for any $\alpha \in (0,1]$, we find a sufficient condition on the probability distribution of $Y$ for the achievement of the optimal rate of convergence, that is $1/n$. Our main result improve on existing literature: in particular, with respect to \cite{MPS}, we study a stronger metric while, with respect to \cite{Mna}, we weaken the regularity hypothesis on the probability distribution of $Y$.