 
Summary: Submitted to the Annals of Statistics
arXiv: math.PR/0911.1164
KERNEL ESTIMATORS OF ASYMPTOTIC VARIANCE
FOR ADAPTIVE MARKOV CHAIN MONTE CARLO
By Yves F. Atchad´e
University of Michigan
We study the asymptotic behavior of kernel estimators of asymp
totic variances (or longrun variances) for a class of adaptive Markov
chains. The convergence is studied both in Lp
and almost surely. The
results apply to Markov chains as well and improve on the existing lit
erature by imposing weaker conditions. We illustrate the results with
applications to the GARCH(1, 1) Markov model and to an adaptive
MCMC algorithm for Bayesian logistic regression.
1. Introduction. Adaptive Markov Chain Monte Carlo (Adaptive MCMC)
provides a flexible framework to optimizing MCMC samplers on the fly
(see e.g. [27, 3, 8] and the reference therein). If is the probability mea
sure of interest, these adaptive MCMC samplers generate random processes
{Xn, n 0} that typically are not Markov but nevertheless satisfy a law
of large numbers and the empirical average n1 n
