• Bernoulli
  • Volume 15, Number 4 (2009), 1351-1367.

Rate of convergence of predictive distributions for dependent data

Patrizia Berti, Irene Crimaldi, Luca Pratelli, and Pietro Rigo

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This paper deals with empirical processes of the type $$C_{n}(B)=\sqrt{n}\{\mu_{n}(B)-P(X_{n+1}\in B\mid X_{1},\ldots,X_{n})\},$$ where $(X_n)$ is a sequence of random variables and $μ_n=(1/n)∑_{i=1}^nδ_{X_i}$ the empirical measure. Conditions for $\sup_B|C_n(B)|$ to converge stably (in particular, in distribution) are given, where $B$ ranges over a suitable class of measurable sets. These conditions apply when $(X_n)$ is exchangeable or, more generally, conditionally identically distributed (in the sense of Berti et al. [Ann. Probab. 32 (2004) 2029–2052]). By such conditions, in some relevant situations, one obtains that $\sup_{B}|C_{n}(B)|\stackrel{P}{\rightarrow}0$ or even that $\sqrt{n}\sup_{B}|C_{n}(B)|$ converges a.s. Results of this type are useful in Bayesian statistics.

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Bernoulli, Volume 15, Number 4 (2009), 1351-1367.

First available in Project Euclid: 8 January 2010

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Bayesian predictive inference central limit theorem conditional identity in distribution empirical distribution exchangeability predictive distribution stable convergence


Berti, Patrizia; Crimaldi, Irene; Pratelli, Luca; Rigo, Pietro. Rate of convergence of predictive distributions for dependent data. Bernoulli 15 (2009), no. 4, 1351--1367. doi:10.3150/09-BEJ191.

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