Kernel estimators of asymptotic variance for adaptive Markov chain Monte Carlo

TL;DR

This paper investigates kernel estimators for asymptotic variances in adaptive Markov chains, establishing convergence under weaker conditions and demonstrating applications in GARCH models and Bayesian logistic regression.

Contribution

It introduces new convergence results for kernel estimators of asymptotic variances in adaptive Markov chains under weaker assumptions.

Findings

Convergence in $L^p$ and almost surely established.

Applicable to GARCH(1,1) models and adaptive MCMC algorithms.

Improves upon existing literature with weaker conditions.

Abstract

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.