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

Given a sequence {X-n}(n >= 1) of exchangeable Bernoulli random variables, the celebrated de Finetti representation theorem states that 1/n Sigma(n)(i)=1(X)i ->(a.s.) Y for a suitable random variable Y : Omega -> [0, 1] satisfying P[X-1 = x(1), ..., X-n = x(n)[Y] = Y(Sigma)i(n)=1 x(i) (1 - Y)(n)-Sigma(i)(=1)nx(i). In this paper, we study the rate of convergence in law of 1/n Sigma(i)(=1)nX(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 is an element of (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.