On the rate of convergence in de Finetti’s representation theorem

Abstract

A consequence of de Finetti's representation theorem is that for every infinite sequence of exchangeable 0-1 random variables (X k ) k≥1 , there exists a probability measure µ on the Borel sets of [0, 1] such thatX n = n −1 n i=1 X i converges weakly to µ. For a wide class of probability measures µ having smooth density on (0, 1), we give bounds of order 1/n with explicit constants for the Wasserstein distance between the law ofX n and µ. This extends a recent result by Goldstein and Reinert [10] regarding the distance between the scaled number of white balls drawn in a Pólya-Eggenberger urn and its limiting distribution. We prove also that, in the most general cases, the distance between the law ofX n and µ is bounded below by 1/n and above by 1/ √ n (up to some multiplicative constants). For every δ ∈ [1/2, 1], we give an example of an exchangeable sequence such that this distance is of order 1/n δ .

Asymptotic predictive inference with exchangeable data

“…Define π n (C) = P µ n {1} ∈ C and π * n (C) = P a n {1} ∈ C for C ∈ C. Because of Theorem 7, n µ n {1} − a n {1} converges a.s. whenever the distribution of µ{1} is absolutely continuous with an almost Lipschitz density f . Our last result, inspired by [22], provides a sharp estimate of ρ(π n , π * n ) under the assumption that f is Lipschitz (and not only almost Lipschitz). Proof.…”

Rate of convergence of empirical measures for exchangeable sequences

“…Define π n (C) = P µ n {1} ∈ C and π * n (C) = P a n {1} ∈ C for C ∈ C. Because of Theorem 7, n µ n {1} − a n {1} converges a.s. whenever the distribution of µ{1} is absolutely continuous with an almost Lipschitz density f . Our last result, inspired by [22], provides a sharp estimate of ρ(π n , π * n ) under the assumption that f is Lipschitz (and not only almost Lipschitz). Then, |X n − E Q (V | G n )| ≤ 1/(n + 2).…”

“…We finally mention a result (Theorem 10) which, though in the spirit of this paper, is quite different from those described above. Such a result has been inspired by [22]. Let S = {0, 1} and C the Borel σ-field on [0, 1].…”

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

“…Despite the long history of the celebrated de Finetti representation theorem, the study of the rate of convergence of μ n to μ has been initiated very recently in the work of Mijoule, Peccati and Swan [19]. In particular, they proved a quantitative version of de Finetti's law of large numbers with respect to the Kantorovich distance (or Wasserstein distance of order 1) d W (ν 1 ; ν 2 ) := 1 0 |F 1 (x) − F 2 (x)| dx.…”