  Given a discrete-valued sample $X_1,...,X_n$ we wish to decide whether it was
generated by a distribution belonging to a family $H_0$, or it was generated by
a distribution belonging to a family $H_1$. 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 $\epsilon$ there exist a sample size $n$
such that probability of error is upper-bounded by $\epsilon$ for samples
longer than $n$. We find some necessary and some sufficient conditions on $H_0$
and $H_1$ under which a consistent test (with this notion of consistency)
exists. These conditions are topological, with respect to the topology of
distributional distance.
