Exact non-parametric sequential convergence test for samplers

Random number generators are extensively used in science. Generating pseudo-random numbers is the base for many data analysis techniques in computational statistics. This is the case, for instance, of most of the Bayesian methods, which are enabled by means of samplers such as the well-known Gibbs sampler and the Metropolis–Hastings. These classical Markov Chain Monte Carlo samplers are designed to generate a sequence of numbers that, under certain conditions, converge to a sequence that behaves as if sampled from a user-defined target distribution. In general, the number of iterations required to reach such convergence is not deterministic. There are several statistical tests for identifying that convergence has not yet been achieved, but not for actually signaling convergence. The present work introduces an exact non-parametric sequential test for signaling the convergence of random number generators in general. The solution is derived in the light of the type I error probability spending approach.