Kernel estimators of asymptotic variance for adaptive Markov chain Monte Carlo

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 Lp 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 GARCH(1,.1) Markov model and to an adaptive MCMC algorithm for Bayesian logistic regression.