Markov Chain Monte Carlo confidence intervals
  For a reversible and ergodic Markov chain $\{X_n,n\geq0\}$ with invariant distribution $\pi$, we show that a valid confidence interval for $\pi(h)$ can be constructed whenever the asymptotic variance $\sigma^2_P(h)$ 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 $\sigma_P^2(h)$. 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. 
Yves F. Atchad\'e
math.ST math.PR stat.TH