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July, 1992 Large Deviations for Exchangeable Random Vectors
I. H. Dinwoodie, S. L. Zabell
Ann. Probab. 20(3): 1147-1166 (July, 1992). DOI: 10.1214/aop/1176989683


Say that a family $\{P_\theta^n: \theta \in \Theta\}$ of sequences of probability measures is exponentially continuous if whenever $\theta_n \rightarrow \theta$, the sequence $\{P_{\theta_n}^n\}$ satisfies a large deviation principle with rate function $\lambda_\theta$. If $\Theta$ is compact and $\{P_\theta^n\}$ is exponentially continuous, then the mixture $P^n(A) =: \int_\Theta P_\theta^n(A)d\mu(\theta)$ satisfies a large deviation principle with rate function $\lambda(x) =: \inf\{\lambda_\theta(x): \theta \in S(\mu)\}$, where $S(\mu)$ is the support of the mixing measure $\mu$. If $X_1,X_2,\ldots$ is a sequence of i.i.d. random vectors, $\{\bar{X}_n\}$ the corresponding sequence of sample means and $P_\theta^n =: P_\theta\circ\bar{X}^{-1}_n$, then $\{P_\theta^n\}$ is exponentially continuous if the classical rate function $\lambda_\theta(\nu)$ is jointly lower semicontinuous and a uniform integrability condition introduced by de Acosta is satisfied. These results are applied in Section 4 to derive a large deviation theory for exchangeable random variables; the resulting rate functions are typically nonconvex. If the parameter space $\Theta$ is not compact, then examples can be constructed where a full large deviation principle is not satisfied because the upper bound fails for a noncompact set.


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I. H. Dinwoodie. S. L. Zabell. "Large Deviations for Exchangeable Random Vectors." Ann. Probab. 20 (3) 1147 - 1166, July, 1992.


Published: July, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0760.60025
MathSciNet: MR1175254
Digital Object Identifier: 10.1214/aop/1176989683

Primary: 60F10
Secondary: 62F20

Keywords: Exchangeable random variables , large deviations , mixtures

Rights: Copyright © 1992 Institute of Mathematical Statistics


Vol.20 • No. 3 • July, 1992
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