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

Given a sequence {Xn} n≥1 of exchangeable Bernoulli random variables, the celebrated de Finetti representation theorem states that 1 n Σ;n i=1 Xi -a→.s. Y for a suitable random variable Y : Ω → [0, 1] satisfying P[X1 = x1, . , Xn = xn|Y] = Y Σn i=1 xi (1 - Y)n- Σn i=1 xi . In this paper, we study the rate of convergence in law of 1 n Σn i=1 Xi to Y under the Kolmogorov distance. After showing that a rate of the type of 1/nα can be obtained for any index α ∈ (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.