  We study the asymptotic behavior of kernel estimators of asymptotic variances
(or long-run variances) for a class of adaptive Markov chains. The convergence
is studied both in $L^p$ and almost surely. The results also apply to Markov
chains and improve on the existing literature by imposing weaker conditions. We
illustrate the results with applications to the $\operatorname {GARCH}(1,1)$
Markov model and to an adaptive MCMC algorithm for Bayesian logistic
regression.
