 
Summary: LIMIT THEOREMS FOR SOME ADAPTIVE MCMC ALGORITHMS
WITH SUBGEOMETRIC KERNELS: PART II
YVES F. ATCHAD´E AND GERSENDE FORT
Aug. 2009
Abstract. We prove a central limit theorem for a general class of adaptive Markov
Chain Monte Carlo algorithms driven by subgeometrically ergodic Markov kernels. We
discuss in detail the special case of stochastic approximation. We use the result to analyze
the asymptotic behavior of an adaptive version of the Metropolis Adjusted Langevin
algorithm with a heavy tailed target density.
1. Introduction
This work is a sequel of Atchade and Fort (2008) and develops central limit theorems
for adaptive MCMC (AMCMC) algorithms. Previous works on the subject include An
drieu and Moulines (2006) and Saksman and Vihola (2009) where central limit theorems
are proved for certain AMCMC algorithms driven by geometrically ergodic Markov ker
nels. There is a need to understand the subgeometric case. Indeed, many Markov kernels
routinely used in practice are not geometrically ergodic. For example, if the target distri
bution of interest has heavy tails, then the Random Walk Metropolis algorithm (RWMA)
and the Metropolis Adjusted Langevin algorithm (MALA) result in subgeometric Markov
kernels (Jarner and Roberts (2002a)).
We consider adaptive MCMC algorithms driven by Markov kernels {P, } such
