Large Deviations for Exchangeable Random Vectors

Abstract

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.