# Large deviations for mixtures

by J.D.Biggins

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

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