An automated stopping rule for MCMC convergence assessment

in this paper, we propose a methodology essentially based on the Central Limit Theorem for Markov chains to monitor convergence of MCMC algorithms using actual outputs. Our methods are grounded on the fact that normality is a testable implication of sufficient mixing. The first control tool tests the normality hypothesis for normalized averages of functions of the Markov chain over independent parallel chains started from a dispersed distribution. A second connected tool is based on graphical monitoring of the stabilization of the variance after n iterations near the limiting variance. Both methods work without knowledge on the sampler driving the chain, and the normality diagnostic leads to automated stopping rules. These stopping rules are implemented in a software toolbox whose performances are illustrated through simulations for finite and continuous state chains reflecting some typical situations and a full scale application. Comparisons are made with the binary control method of Raftery and Lewis.