Rates of convergence in de Finetti’s representation theorem, and Hausdorff moment problem

Abstract

Given a sequence $\{X_{n}\}_{n\geq 1}$ of exchangeable Bernoulli random variables, the celebrated de Finetti representation theorem states that $\frac{1}{n}\sum_{i=1}^{n}X_{i}\stackrel{a.s.}{\longrightarrow }Y$ for a suitable random variable $Y:\Omega \rightarrow [0,1]$ satisfying $\mathsf{P}[X_{1}=x_{1},\dots ,X_{n}=x_{n}|Y]=Y^{\sum_{i=1}^{n}x_{i}}(1-Y)^{n-\sum_{i=1}^{n}x_{i}}$. In this paper, we study the rate of convergence in law of $\frac{1}{n}\sum_{i=1}^{n}X_{i}$ to $Y$ under the Kolmogorov distance. After showing that a rate of the type of $1/n^{\alpha }$ can be obtained for any index $\alpha \in (0,1]$, we find a sufficient condition on the distribution of $Y$ for the achievement of the optimal rate of convergence, that is $1/n$. Besides extending and strengthening recent results under the weaker Wasserstein distance, our main result weakens the regularity hypotheses on $Y$ in the context of the Hausdorff moment problem.