# Large deviations for mixtures

2000 Mathematics Subject Classification: Primary 60F10

## Abstract

Suppose the probability measures $(\mu^n)$ on $\cT$ obey a large deviation principle (LDP). Suppose too that $\mu^n$ is concentrated on $\cT_n$ and that, for $\theta (n) \in \cT_n$ with $\theta (n) \rightarrow \theta \in \cT$, the probability measures $(P^n_{\theta (n)})$ on $\cX$ also obey an LDP. The main purpose of this paper is to give conditions which allow an LDP for the mixtures $(P^n)$, given by $P^n(A)= \int P^n_\theta(A)d \mu^n(\theta)$, to be deduced. Chaganty (1997) also considered this question, but under stronger assumptions. The treatment here follows that of Dinwoodie and Zabell (1992) who, motivated by exchangeability, considered the case where $\mu^n$ does not vary with $n$.

## Availability:

Preprint 536/03, Department of Probability and Statistics, University of Sheffield. (September 2003.) pdf file

Version 2 (pdf) Nov 2003; Final Version (pdf): minor changes from version 2

Electronic Communications in Probability (2004) 9, 60-71

Other publications by J.D. Biggins