 
Summary: To appear Annals of Applied Probability
2005Vol. 0, No. 0, 18
ON THE ERGODICITY PROPERTIES OF SOME ADAPTIVE MCMC
ALGORITHMS
By Christophe Andrieu, and ´Eric Moulines
University of Bristol and ´Ecole Nationale Sup´erieure des T´el´ecommunications
In this paper we study the ergodicity properties of some adap
tive Monte Carlo Markov chain algorithms (MCMC) that have been
recently proposed in the literature. We prove that under a set of ver
ifiable conditions, ergodic averages calculated from the output of a
socalled adaptive MCMC sampler converge to the required value and
can even, under more stringent assumptions, satisfy a central limit
theorem. We prove that the conditions required are satisfied for the
Independent MetropolisHastings algorithm and the Random Walk
Metropolis algorithm with symmetric increments. Finally we propose
an application of these results to the case where the proposal distrib
ution of the MetropolisHastings update is a mixture of distributions
from a curved exponential family.
1. Introduction. Markov chain Monte Carlo (MCMC), introduced by Metropolis
et al. (1953), is a popular computational method for generating samples from virtually any
