A criterion for hypothesis testing for stationary processes

Given a finite-valued sample $X_1,...,X_n$ we wish to test whether it was generated by a stationary ergodic process belonging to a family $H_0$, or it was generated by a stationary ergodic process outside $H_0$. We require the Type I error of the test to be uniformly bounded, while the type II error has to be mande not more than a finite number of times with probability 1. For this notion of consistency we provide necessary and sufficient conditions on the family $H_0$ for the existence of a consistent test. This criterion is illustrated with applications to testing for a membership to parametric families, generalizing some existing results. In addition, we analyze a stronger notion of consistency, which requires finite-sample guarantees on error of both types, and provide some necessary and some sufficient conditions for the existence of a consistent test. We emphasize that no assumption on the process distributions are made beyond stationarity and ergodicity.