 
Summary: On the Geometric Ergodicity of MetropolisHastings
Algorithms
Yves F. Atchadé1 and François Perron2
(March, 2003; revised August, 2005)
Abstract
Under a compactness assumption, we show that a irreducible and aperiodic Metropolis
Hastings chain is geometrically ergodic if and only if its rejection probability is bounded
away from unity. In the particular case of the Independence MetropolisHastings algorithm,
we obtain that the whole spectrum of the induced operator is contained in (and in many
cases equal to) the essential range of the rejection probability of the chain as conjectured
by Liu (1996).
Key words: Geometric ergodicity, Markov chain operators, MetropolisHastings algorithm.
MSC Numbers: 65C05, 65C40, 60J27, 60J35
1 Introduction
The MetropolisHastings (MH) algorithm is a very exible algorithm used to approximately sample
from complicated distributions in high dimension spaces. If is the probability distribution of inter
est, such an algorithm generates a Markov chain (Xn) which admits as its stationary distribution.
Geometric ergodicity characterizes a global stability property of the chain that is particularly useful
from a statistical point of view. For example, if the Markov chain is geometric ergodicity, central
limit theorems for empirical sums of functional of the chain are easier to obtain (see e.g. Jones
