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

Abstract

Given a sequence {X n } n≥1 of exchangeable Bernoulli random variables, the celebrated de Finetti representation theorem states that 1 n n i=1 X i a.s.

A Compound Poisson Perspective of Ewens–Pitman Sampling Model

“…This can be seen in Section 6.1 of Pitman [23] and the references therein. Such a line of research aims to extend the preliminary works of Dolera and Favaro [16,17] on Berry-Esseen type theorems under the EP-SM. Work on this, and on the more general settings induced by normalised random measures (Regazzini et al [24]) and Poisson-Kingman models (Pitman [23]), is ongoing.…”

“…Theorem 2 presents a compound Poisson perspective of the EP-SM in terms of the NB-CPSM, thus extending the well-known compound Poisson perspective of the E-SM in terms of the LS-CPSM. Statement (i) of Theorem 2 shows that for α ∈ (0, 1) and θ > −α, the EP-SM admits a representation in terms of the NB-CPSM with α ∈ (0, 1) and z > 0, where the randomisation acts on the parameter z with respect to the distribution (17). Precisely, this is a compound mixed Poisson sampling model.…”

“…That is, a compound sampling model in which the distribution of the random number K of distinct types in the population is a mixture of Poisson distributions with respect to the law of Xα,θ,n . Statement (ii) of Theorem 2 shows that for α < 0 and z < 0, the NB-CPSM admits a representation in terms of a randomised EP-SM with α < 0 and θ = −mα for some m ∈ N, where the randomisation acts on the parameter m with respect to the distribution (17).…”

“…. , M n,n (α, θ)) be distributed as the EP-SM (1) and let Xα,θ,n be the random variable defined in (17), which is independent of (M 1,n (α, θ), . .…”

“…as n → +∞. We refer to Dolera and Favaro [16,17] for Berry-Esseen type refinements of (20) and to Favaro et al [18,19] and Favaro and James [13] for generalisations of ( 20) with applications to Bayesian nonparametrics. This can also be seen in Pitman [5] (Chapter 4) for a general treatment of (20).…”