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May 2020 Rates of convergence in de Finetti’s representation theorem, and Hausdorff moment problem
Emanuele Dolera, Stefano Favaro
Bernoulli 26(2): 1294-1322 (May 2020). DOI: 10.3150/19-BEJ1156

Abstract

Given a sequence $\{X_{n}\}_{n\geq 1}$ of exchangeable Bernoulli random variables, the celebrated de Finetti representation theorem states that $\frac{1}{n}\sum_{i=1}^{n}X_{i}\stackrel{a.s.}{\longrightarrow }Y$ for a suitable random variable $Y:\Omega \rightarrow [0,1]$ satisfying $\mathsf{P}[X_{1}=x_{1},\dots ,X_{n}=x_{n}|Y]=Y^{\sum_{i=1}^{n}x_{i}}(1-Y)^{n-\sum_{i=1}^{n}x_{i}}$. In this paper, we study the rate of convergence in law of $\frac{1}{n}\sum_{i=1}^{n}X_{i}$ to $Y$ under the Kolmogorov distance. After showing that a rate of the type of $1/n^{\alpha }$ can be obtained for any index $\alpha \in (0,1]$, 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.

Citation

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Emanuele Dolera. Stefano Favaro. "Rates of convergence in de Finetti’s representation theorem, and Hausdorff moment problem." Bernoulli 26 (2) 1294 - 1322, May 2020. https://doi.org/10.3150/19-BEJ1156

Information

Received: 1 January 2019; Revised: 1 July 2019; Published: May 2020
First available in Project Euclid: 31 January 2020

zbMATH: 07166564
MathSciNet: MR4058368
Digital Object Identifier: 10.3150/19-BEJ1156

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

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Vol.26 • No. 2 • May 2020
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