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 $Y$.
Categories: math.PR
