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

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 $Y$.