Given a sequence {X_n} of exchangeable Bernoulli random variables, the celebrated de Finetti representation theorem states that (X_1+...+X_n)/n → Y a.s. for a suitable random variable Y : Omega → [0, 1] satisfying P[X_1 = x_1,..., X_n = x_n |Y]=Y^(X_1+...+X_n) * (1−Y)^(n - X_1-...-X_n). In this paper, we study the rate of convergence in law of (X_1+...+X_n)/n to Y under the Kolmogorov distance. After showing that a rate of the type of (1/n)^α 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.
Rates of convergence in de Finetti’s representation theorem, and Hausdorff moment problem
Dolera Emanuele;
2020-01-01
Abstract
Given a sequence {X_n} of exchangeable Bernoulli random variables, the celebrated de Finetti representation theorem states that (X_1+...+X_n)/n → Y a.s. for a suitable random variable Y : Omega → [0, 1] satisfying P[X_1 = x_1,..., X_n = x_n |Y]=Y^(X_1+...+X_n) * (1−Y)^(n - X_1-...-X_n). In this paper, we study the rate of convergence in law of (X_1+...+X_n)/n to Y under the Kolmogorov distance. After showing that a rate of the type of (1/n)^α 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
