Uniform hypothesis testing for ergodic time series distributions

Given a discrete-valued sample X₁, ..., X_n we wish to decide whether it was generated by a distribution belonging to a family H₀, or it was generated by a distribution belonging to a family H₁. In this work we assume that all distributions are stationary ergodic, and do not make any further assumptions (e.g. no independence or mixing rate assumptions). We would like to have a test whose probability of error (both Type I and Type II) is uniformly bounded. More precisely, we require that for each ε there exist a sample size n such that probability of error is upper-bounded by ε for samples longer than n. We find some necessary and some sufficient conditions on H₀ and H₁ under which a consistent test (with this notion of consistency) exists. These conditions are topological, with respect to the topology of distributional distance.