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November 2018 Asymptotic predictive inference with exchangeable data
Patrizia Berti, Luca Pratelli, Pietro Rigo
Braz. J. Probab. Stat. 32(4): 815-833 (November 2018). DOI: 10.1214/17-BJPS367


Let $(X_{n})$ be a sequence of random variables, adapted to a filtration $(\mathcal{G}_{n})$, and let $\mu_{n}=(1/n)\sum_{i=1}^{n}\delta_{X_{i}}$ and $a_{n}(\cdot)=P(X_{n+1}\in\cdot|\mathcal{G}_{n})$ be the empirical and the predictive measures. We focus on \begin{equation*}\Vert \mu_{n}-a_{n}\Vert =\mathop{\mathrm{sup}}_{B\in\mathcal{D}}\vert\mu_{n}(B)-a_{n}(B)\vert,\end{equation*} where $\mathcal{D}$ is a class of measurable sets. Conditions for $\Vert \mu_{n}-a_{n}\Vert \rightarrow0$, almost surely or in probability, are given. Also, to determine the rate of convergence, the asymptotic behavior of $r_{n}\Vert \mu_{n}-a_{n}\Vert $ is investigated for suitable constants $r_{n}$. Special attention is paid to $r_{n}=\sqrt{n}$ and $r_{n}=\sqrt{\frac{n}{\log\log n}}$. The sequence $(X_{n})$ is exchangeable or, more generally, conditionally identically distributed.


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Patrizia Berti. Luca Pratelli. Pietro Rigo. "Asymptotic predictive inference with exchangeable data." Braz. J. Probab. Stat. 32 (4) 815 - 833, November 2018.


Received: 1 March 2016; Accepted: 1 May 2017; Published: November 2018
First available in Project Euclid: 17 August 2018

zbMATH: 06979602
MathSciNet: MR3845031
Digital Object Identifier: 10.1214/17-BJPS367

Keywords: Bayesian consistency , conditional identity in distribution , empirical measure , exchangeability , predictive measure , random probability measure

Rights: Copyright © 2018 Brazilian Statistical Association


Vol.32 • No. 4 • November 2018
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