 
Summary: LIMIT THEOREMS FOR SOME ADAPTIVE MCMC ALGORITHMS
WITH SUBGEOMETRIC KERNELS
YVES ATCHAD´E AND GERSENDE FORT
Abstract. This paper deals with the ergodicity (convergence of the marginals) and
the law of large numbers for adaptive MCMC algorithms built from transition kernels
that are not necessarily geometrically ergodic. We develop a number of results that
broaden significantly the class of adaptive MCMC algorithms for which rigorous analysis
is now possible. As an example, we give a detailed analysis of the Adaptive Metropolis
Algorithm of Haario et al. (2001) when the target distribution is subexponential in the
tails.
1. Introduction
This paper deals with the convergence of Adaptive Markov Chain Monte Carlo (AM
CMC). Markov Chain Monte Carlo (MCMC) is a well known, widely used method to sam
ple from arbitrary probability distributions. One of the major limitation of the method is
the difficulty in finding sensible values for the parameters of the Markov kernels. Adap
tive MCMC provides a general framework to tackle this problem where the parameters
are adaptively tuned, often using previously generated samples. This approach generates
a class of stochastic processes that is the object of this paper.
Denote the probability measure of interest on some measure space (X, X). Let {P,
} be a family of irreducible and aperiodic Markov kernels each with invariant distribu
