Markov Chain Monte Carlo Confidence Intervals

For a reversible and ergodic Markov chain {Xn, n ≥ 0} with invariant distribution π, we show that a valid confidence interval for π(h) can be constructed whenever the asymptotic variance σ P (h) is finite and positive. We do not impose any additional condition on the convergence rate of the Markov chain. The confidence interval is derived using the so-called fixed-b lag-window estimator of σ P (h). We also derive a result that suggests that the proposed confidence interval procedure converges faster than classical confidence interval procedures based on the Gaussian distribution and standard central limit theorems for Markov chains.